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An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov... 1IntroductionFourth-order Steklov eigenvalue problems with eigenvalue parameter in boundary conditions are widely used in mathematics and physics, such as the surface wave research, the stability analysis of mechanical oscillator in a viscous fluid, the study of vibration mode of structure in contact with an incompressible fluid, and so on [1,2,3, 4,5]. The first eigenvalue λ1{\lambda }_{1}also plays an important role in the positivity-preserving properties for the biharmonic-operator Δ2{\Delta }^{2}under the boundary conditions ϕ∣∂Ω=(Δϕ−λϕν)∣∂Ω=0\phi \hspace{-0.25em}{| }_{\partial \Omega }=\left(\Delta \phi -\lambda {\phi }_{\nu }){| }_{\partial \Omega }=0in [6,7].There are many existing results about the fourth-order Steklov eigenvalue problems, but they mainly focus on the qualitative analysis. Kuttler [8] proved that the first eigenvalue is simple and the corresponding eigenfunction does not change the sign. Ferrero et al. [7] and Bucur et al. [9] studied the spectrum on a bounded domain, and the explicit representation of the spectrum is given when the domain is a ball. Recently, the existence of an optimal convex shape among domains of a given measure is proved in [10], and the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues also is established in [11]. For the numerical methods of the fourth-order Steklov eigenvalue problems, a conforming finite element method was first proposed in [12], then some spectral methods are also developed [13].As we all know, if the conforming finite element method is directly used to solve a fourth-order problem, the boundary of the element requires the continuity of the first derivative, which not only brings the difficulty of constructing the basis function but also costs a lot of calculation time and memory capacity, especially for some special regions, such as circular region, spherical region, and so on. How to efficiently solve a fourth-order Steklov eigenvalue problem in a circular domain? To the best of our knowledge, there are few reports on using some efficient numerical to solve this problem. Thus, the aim of this article is to propose an effective finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.This article is organized as follows. In Section 2, a reduced scheme based on polar coordinate transformation is presented. In Section 3, the weighted space and discrete variational form are derived. In Section 4, the error estimation of approximation solutions is proved. In Section 5, we present the process of effective implementation of the algorithm. We present some numerical experiments in Section 6 to illustrate the accuracy and efficiency of our proposed algorithm. Finally, we give in Section 7 some concluding remarks.2Reduced scheme based on polar coordinate transformationThe fourth-order Steklov eigenvalue problems read: (2.1)Δ2ϕ=0,inΩ,{\Delta }^{2}\phi =0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,(2.2)ϕ=0,on∂Ω,\phi =0,\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,(2.3)Δϕ=λ∂ϕ∂ν,on∂Ω,\Delta \phi =\lambda \frac{\partial \phi }{\partial \nu },\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,where Ω={(x,y)∈R2:0<x2+y2<R}\Omega =\left\{\left(x,y)\in {{\mathbb{R}}}^{2}:0\lt \sqrt{{x}^{2}+{y}^{2}}\lt R\right\}, ν\nu is the unit outward normal to the boundary ∂Ω\partial \Omega . Let x=rcosθ,y=rsinθx=r\cos \theta ,y=r\sin \theta , ψ(r,θ)=ϕ(x,y)\psi \left(r,\theta )=\phi \left(x,y). Then we derive that (2.4)Δϕ(x,y)=Lψ(r,θ)=1r∂∂rr∂ψ(r,θ)∂r+1r2∂2ψ(r,θ)∂θ2.\Delta \phi \left(x,y)=L\psi \left(r,\theta )=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \psi \left(r,\theta )}{\partial r}\right)+\frac{1}{{r}^{2}}\frac{{\partial }^{2}\psi \left(r,\theta )}{\partial {\theta }^{2}}.Then the equivalent form of (2.1)–(2.3) in polar coordinates is as follows: (2.5)L2ψ(r,θ)=0,inD=(0,R)×[0,2π),{L}^{2}\psi \left(r,\theta )=0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathcal{D}}=\left(0,R)\times \left[0,2\pi ),(2.6)ψ(R,θ)=0,θ∈[0,2π),\psi \left(R,\theta )=0,\hspace{1em}\theta \in \left[0,2\pi ),(2.7)Lψ(R,θ)=λ∂ψ∂r(R,θ),θ∈[0,2π).L\psi \left(R,\theta )=\lambda \frac{\partial \psi }{\partial r}\left(R,\theta ),\hspace{1em}\theta \in \left[0,2\pi ).\hspace{1.75em}Since ψ(r,θ)\psi \left(r,\theta )is 2π2\pi periodic in θ\theta , then we have (2.8)ψ(r,θ)=∑∣m∣=0∞ψm(r)eimθ.\psi \left(r,\theta )=\mathop{\sum }\limits_{| m| =0}^{\infty }{\psi }_{m}\left(r){{\rm{e}}}^{im\theta }.Substituting (2.8) into (2.4), we derive that (2.9)Lψ(r,θ)=∑∣m∣=0∞1r∂∂rr∂ψm(r)∂r−m2rψm(r)eimθ.L\psi \left(r,\theta )=\mathop{\sum }\limits_{| m| =0}^{\infty }\frac{1}{r}\left\{\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]-\frac{{m}^{2}}{r}{\psi }_{m}\left(r)\right\}{{\rm{e}}}^{im\theta }.Following the discussion in [14,15], to overcome the pole singularity introduced by polar coordinate transformation, we need to introduce the essential pole conditions, which make (2.9) meaningful, as follows: (2.10)m2ψm(0)=0,limr→0+∂∂rr∂ψm(r)∂r−m2rψm(r)=0.{m}^{2}{\psi }_{m}\left(0)=0,\mathop{\mathrm{lim}}\limits_{r\to {0}^{+}}\left\{\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]-\frac{{m}^{2}}{r}{\psi }_{m}\left(r)\right\}=0.Using the fact that ∂∂rr∂ψm(r)∂r=∂ψm(r)∂r+r∂2ψm(r)∂r2\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]=\frac{\partial {\psi }_{m}\left(r)}{\partial r}+r\frac{{\partial }^{2}{\psi }_{m}\left(r)}{\partial {r}^{2}}, (2.10) can be reduced to (2.11)m2ψm(0)=0,(1−m2)∂ψm(0)∂r=0.{m}^{2}{\psi }_{m}\left(0)=0,\hspace{1em}\left(1-{m}^{2})\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0.From (2.11) we can further obtain that (2.12)(1)∂ψm(0)∂r=0,(m=0);\left(1)\hspace{1em}\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0,\hspace{1em}\left(m=0);\hspace{6.25em}(2.13)(2)ψm(0)=0,(∣m∣=1);\left(2)\hspace{1em}{\psi }_{m}\left(0)=0,\hspace{1em}\left(| m| =1);\hspace{6.65em}(2.14)(3)ψm(0)=0,∂ψm(0)∂r=0,(∣m∣≥2).\left(3)\hspace{1em}{\psi }_{m}\left(0)=0,\hspace{1em}\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0,\hspace{1em}\left(| m| \ge 2).Let r=t+12R,um(t)=ψm(r),Lmum(t)=1t+1∂∂tt∂um(t)∂t−m2(t+1)2um(t)r=\frac{t+1}{2}R,{u}_{m}\left(t)={\psi }_{m}\left(r),{L}_{m}{u}_{m}\left(t)=\frac{1}{t+1}\frac{\partial }{\partial t}\left[t\frac{\partial {u}_{m}\left(t)}{\partial t}\right]-\frac{{m}^{2}}{{\left(t+1)}^{2}}{u}_{m}\left(t). From (2.8), (2.11)–(2.14), and the orthogonal properties of Fourier basis functions, (2.5)–(2.7) are equivalent to a series of one-dimensional eigenvalue problems (2.15)Lm2um(t)=0,t∈(−1,1),{L}_{m}^{2}{u}_{m}\left(t)=0,\hspace{1em}t\in \left(-1,1),(2.16)(1)∂um(−1)∂t=0,um(1)=0,Lmum(1)=R2λm∂um(1)∂t,(m=0);\left(1)\hspace{1em}\frac{\partial {u}_{m}\left(-1)}{\partial t}=0,\hspace{1em}{u}_{m}\left(1)=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(m=0);\hspace{.1em}(2.17)(2)um(−1)=0,um(1)=0,Lmum(1)=R2λm∂um(1)∂t,(∣m∣=1);\left(2)\hspace{1em}{u}_{m}\left(-1)=0,\hspace{1em}{u}_{m}\left(1)=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(| m| =1);\hspace{.5em}(2.18)(3)um(±1)=0,∂um(−1)∂t=0,Lmum(1)=R2λm∂um(1)∂t,(∣m∣≥2).\left(3)\hspace{1em}{u}_{m}\left(\pm 1)=0,\hspace{1em}\frac{\partial {u}_{m}\left(-1)}{\partial t}=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(| m| \ge 2).3Weighted space and discrete variational formWithout losing generality, we only consider the case of m≥0m\ge 0. First, we divide the solution interval I=(−1,1)I=\left(-1,1)as follows: −1=t0<t1<⋯<ti<⋯<tn=1.-1={t}_{0}\lt {t}_{1}\hspace{0.33em}\lt \cdots \lt {t}_{i}\hspace{0.33em}\lt \cdots \lt {t}_{n}=1.Define the usual weighted Sobolev space: Lω2(I)≔ρ:∫Iωρ2dt<∞{L}_{\omega }^{2}\left(I):= \left\{\rho :\mathop{\int }\limits_{I}\omega {\rho }^{2}{\rm{d}}t\lt \infty \right\}equipped with the following inner product and norm: (ρ,v)ω=∫Iωρvdt,‖ρ‖w=∫Iωρ2dt12,{\left(\rho ,v)}_{\omega }=\mathop{\int }\limits_{I}\omega \rho v{\rm{d}}t,\hspace{1em}\Vert \rho {\Vert }_{w}={\left(\mathop{\int }\limits_{I}\omega {\rho }^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}},where ω=1+t,t∈(−1,1)\omega =1+t,t\in \left(-1,1). We further introduce the following weighted Sobolev space: H0,ω,m2(I)≔um:Lmum∈Lω2(I),m2um(−1)=(1−m2)∂um(−1)∂t=um(1)=0,{H}_{0,\omega ,m}^{2}\left(I):= \left\{{u}_{m}:{L}_{m}{u}_{m}\in {L}_{\omega }^{2}\left(I),{m}^{2}{u}_{m}\left(-1)=\left(1-{m}^{2})\frac{\partial {u}_{m}\left(-1)}{\partial t}={u}_{m}\left(1)=0\right\},equipped with the inner product and norm: (um,vm)2,ω,m=(Lmum,Lmvm)ω,‖um‖2,ω,m=(um,um)2,ω,m.{\left({u}_{m},{v}_{m})}_{2,\omega ,m}={\left({L}_{m}{u}_{m},{L}_{m}{v}_{m})}_{\omega },\Vert {u}_{m}{\Vert }_{2,\omega ,m}=\sqrt{{\left({u}_{m},{u}_{m})}_{2,\omega ,m}}.Then the variational form of (2.15)–(2.18) is: Find (λm,um≠0)∈R×H0,ω,m2(I)\left({\lambda }_{m},{u}_{m}\ne 0)\in {\mathbb{R}}\times {H}_{0,\omega ,m}^{2}\left(I), such that (3.1)Am(um,vm)=λmBm(um,vm),∀vm∈H0,ω,m2(I),{A}_{m}\left({u}_{m},{v}_{m})={\lambda }_{m}{B}_{m}\left({u}_{m},{v}_{m}),\hspace{1em}\forall {v}_{m}\in {H}_{0,\omega ,m}^{2}\left(I),where Am(um,vm)=∫I(t+1)LmumLmvmdt,Bm(um,vm)=R∂um(1)∂t∂vm(1)∂t.\begin{array}{rcl}{A}_{m}\left({u}_{m},{v}_{m})& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t,\\ {B}_{m}\left({u}_{m},{v}_{m})& =& R\frac{\partial {u}_{m}\left(1)}{\partial t}\frac{\partial {v}_{m}\left(1)}{\partial t}.\end{array}Let us denote by Uh{U}_{h}a piecewise cubic Hermite interpolation function space. Define the approximation space Sh(m)=Uh∩H0,ω,m2(I){S}_{h}\left(m)={U}_{h}\cap {H}_{0,\omega ,m}^{2}\left(I). Then the discrete variational form associated with (3.1) is: Find (λmh,umh≠0)∈R×Sh(m)\left({\lambda }_{mh},{u}_{mh}\ne 0)\in {\mathbb{R}}\times {S}_{h}\left(m), such that (3.2)Am(umh,vmh)=λmhBm(umh,vmh),∀vmh∈Sh(m).{A}_{m}\left({u}_{mh},{v}_{mh})={\lambda }_{mh}{B}_{m}\left({u}_{mh},{v}_{mh}),\hspace{1em}\forall {v}_{mh}\in {S}_{h}\left(m).4Error estimation of approximation solutionsFor the sake of brevity, we shall use the expression a≲ba\lesssim bwhich denotes a≤cba\le cb, where ccis a positive constant.Lemma 1For any um,vm∈H0,ω,m2(I){u}_{m},{v}_{m}\in {H}_{0,\omega ,m}^{2}\left(I), the following equalities hold: (4.1)∫I(t+1)LmumLmvmdt=∫I(t+1)um″vm″dt+(2m2+1)∫I1t+1um′vm′dt+m2(m2−4)∫I1(t+1)3umvmdt+um′(1)vm′(1)\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\left(2{m}^{2}+1)\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t\\ & & +{m}^{2}\left({m}^{2}-4)\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+{u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1)\end{array}with m≠1m\ne 1, and(4.2)∫I(t+1)LmumLmvmdt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1umvm′−mt+1vmdt+(1+m)um′(1)vm′(1)\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t\\ & & +{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m){u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1)\end{array}with m=1m=1.ProofUsing integration by parts, pole conditions, and boundary conditions, we derive that (4.3)∫I(um″vm′+um′vm″)dx=um′(1)vm′(1),\mathop{\int }\limits_{I}\left({u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime} }+{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime\prime} }){\rm{d}}x={u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1),(4.4)∫I1(t+1)2(um′vm+umvm′)dt=2∫I1(t+1)3umvmdt,\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}\left({u}_{m}^{^{\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime} }){\rm{d}}t=2\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t,(4.5)∫I1t+1(um″vm+umvm″)dt=2∫I1(t+1)3umvmdt−2∫I1(t+1)um′vm′dt.\mathop{\int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime\prime} }){\rm{d}}t=2\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t-2\mathop{\int }\limits_{I}\frac{1}{\left(t+1)}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t.Then when m≠1m\ne 1, we derive from (4.3)–(4.5) that ∫I(t+1)LmumLmvmdt=∫I(t+1)um″+um′−m2t+1umvm″+vm′t+1−m2(t+1)2vmdt=∫I(t+1)um″vm″dt+∫Ium″vm′+um′vm″dt−m2∫I1(t+1)2(um′vm+umvm′)dt−m2∫I1t+1(um″vm+umvm″)dt+m4∫I1(t+1)3umvmdt+∫I1(t+1)um′vm′dt=∫I(t+1)um″vm″dt+(2m2+1)∫I1t+1um′vm′dt+m2(m2−4)∫I1(t+1)3umvmdt+um′(1)vm′(1).\hspace{-1em}\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left[\left(t+1){u}_{m}^{^{\prime\prime} }+{u}_{m}^{^{\prime} }-\frac{{m}^{2}}{t+1}{u}_{m}\right]\left[{v}_{m}^{^{\prime\prime} }+\frac{{v}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{v}_{m}\right]{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}{u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime} }+{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t-{m}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}\left({u}_{m}^{^{\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime} }){\rm{d}}t\\ & & -{m}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime\prime} }){\rm{d}}t+{m}^{4}\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{\left(t+1)}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\left(2{m}^{2}+1)\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t+{m}^{2}\left({m}^{2}-4)\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+{u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1).\end{array}When m=1m=1, we have ∫I(t+1)LmumLmvmdt=∫I(t+1)um″+um′t+1−m2(t+1)2umvm″+vm′t+1−m2(t+1)2vmdt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1um×vm′−mt+1vmdt+(1+m)∫Ium′−mt+1umvm′−mt+1vm′dt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1um×vm′−mt+1vmdt+(1+m)um′(1)vm′(1).□\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1)\left[{u}_{m}^{^{\prime\prime} }+\frac{{u}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{u}_{m}\right]\left[{v}_{m}^{^{\prime\prime} }+\frac{{v}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{v}_{m}\right]{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t+{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\\ & & \times \left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m)\mathop{\displaystyle \int }\limits_{I}{\left[\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)\right]}^{^{\prime} }{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t+{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\\ & & \times \left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m){u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1).\hspace{18em}\square \end{array}Theorem 1Am(um,vm){A}_{m}\left({u}_{m},{v}_{m})is a bounded and coercive bilinear functional on H0,ω,m2(I)×H0,ω,m2(I){H}_{0,\omega ,m}^{2}\left(I)\times {H}_{0,\omega ,m}^{2}\left(I), i.e., ∣Am(um,vm)∣≲‖um‖2,ω,m‖vm‖2,ω,m,| {A}_{m}\left({u}_{m},{v}_{m})| \lesssim \Vert {u}_{m}{\Vert }_{2,\omega ,m}\Vert {v}_{m}{\Vert }_{2,\omega ,m},Am(um,um)≳‖um‖2,ω,m2.{A}_{m}\left({u}_{m},{u}_{m})\gtrsim \Vert {u}_{m}{\Vert }_{2,\omega ,m}^{2}.ProofFrom Cauchy-Schwarz inequality, we derive that ∣Am(um,vm)∣=∫I(t+1)LmumLmvmdt≤∫I(t+1)∣Lmum∣2dt12∫I(t+1)∣Lmvm∣2dt12≲‖um‖2,ω,m‖v‖2,ω,m,Am(um,um)=∫I(t+1)∣Lmum∣2dt≳‖um‖2,ω,m2.□\hspace{7.8em}\begin{array}{rcl}| {A}_{m}\left({u}_{m},{v}_{m})| & =& \left|\hspace{-0.33em}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t\hspace{-0.33em}\right|\\ & \le & {\left[\mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{u}_{m}{| }^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}{\left[\mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{v}_{m}{| }^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}\\ & \lesssim & \Vert {u}_{m}{\Vert }_{2,\omega ,m}\Vert v{\Vert }_{2,\omega ,m},\\ {A}_{m}\left({u}_{m},{u}_{m})& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{u}_{m}{| }^{2}{\rm{d}}t\gtrsim \Vert {u}_{m}{\Vert }_{2,\omega ,m}^{2}.\hspace{18em}\square \end{array}Lemma 3.2. Let λml{\lambda }_{m}^{l}be the lth eigenvalue of the variational form (3.1). Use Vl{V}_{l}to denote any ll-dimensional subspace of H0,ω,m2(I){H}_{0,\omega ,m}^{2}\left(I). For λm1≤λm2≤⋯≤λml≤⋯{\lambda }_{m}^{1}\le {\lambda }_{m}^{2}\hspace{0.33em}\hspace{0.33em}\le \cdots \le {\lambda }_{m}^{l}\hspace{0.33em}\le \cdots , it holds(4.6)λml=minVl⊂H0,ω,m2(I)maxvm∈VlAm(vm,vm)Bm(vm,vm).{\lambda }_{m}^{l}=\mathop{\min }\limits_{{V}_{l}\subset {H}_{0,\omega ,m}^{2}\left(I)}\mathop{\max }\limits_{{v}_{m}\in {V}_{l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}.ProofSee Theorem 3.1 in [17].□Lemma 3.3. Let λmi{\lambda }_{m}^{i}be the eigenvalue of the variational form (3.1) and be arranged in the ascending order. DefineWi,j=span{umi,…,umj},{W}_{i,j}={\rm{span}}\{{u}_{m}^{i},\ldots ,{u}_{m}^{j}\},where umi{u}_{m}^{i}is the eigenfunction associated with λmi{\lambda }_{m}^{i}. Then there hold(4.7)λml=maxvm∈Wk,lAm(vm,vm)Bm(vm,vm)k≤l,{\lambda }_{m}^{l}=\mathop{\max }\limits_{{v}_{m}\in {W}_{k,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\hspace{1em}k\le l,(4.8)λml=minvm∈Wl,nAm(vm,vm)Bm(vm,vm)l≤n.{\lambda }_{m}^{l}=\mathop{\min }\limits_{{v}_{m}\in {W}_{l,n}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\hspace{1em}l\le n.ProofSee Lemma 3.2 in [17].□For the discrete form (3.2), the following minimax principle is also effective (see [17]).Lemma 3.4. Let λmhl{\lambda }_{mh}^{l}be the eigenvalue of the discrete variational form (3.2). Use Vlh{V}_{lh}to denote any ll-dimensional subspace of Sh(m){S}_{h}\left(m). For λmh1≤λmh2≤⋯≤λmhl≤⋯{\lambda }_{mh}^{1}\le {\lambda }_{mh}^{2}\hspace{0.33em}\le \cdots \le {\lambda }_{mh}^{l}\hspace{0.25em}\le \cdots , it holds(4.9)λmhl=minVhl⊂Sh(m)maxvm∈VhlAm(vm,vm)Bm(vm,vm).{\lambda }_{mh}^{l}=\mathop{\min }\limits_{{V}_{hl}\subset {S}_{h}\left(m)}\mathop{\max }\limits_{{v}_{m}\in {V}_{hl}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}.Define an orthogonal projection Qh2,m:H0,ω,m2(I)→Sh(m){Q}_{h}^{2,m}:{H}_{0,\omega ,m}^{2}\left(I)\to {S}_{h}\left(m)by Am(um−Qh2,mum,vmh)=0,∀vmh∈Sh(m).{A}_{m}\left({u}_{m}-{Q}_{h}^{2,m}{u}_{m},{v}_{mh})=0,\hspace{1em}\forall {v}_{mh}\in {S}_{h}\left(m).Theorem 2Let λmhl{\lambda }_{mh}^{l}be the approximation solution of λml{\lambda }_{m}^{l}. Then it holds(4.10)0<λml≤λmhl≤λmlmaxvm∈W1,lBm(vm,vm)Bm(Qh2,mvm,Qh2,mvm).0\lt {\lambda }_{m}^{l}\le {\lambda }_{mh}^{l}\le {\lambda }_{m}^{l}\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.ProofAccording to the positive definite property of Am(um,vm){A}_{m}\left({u}_{m},{v}_{m})and Bm(um,vm){B}_{m}\left({u}_{m},{v}_{m})we derive that λml>0{\lambda }_{m}^{l}\gt 0. Since Sh(m)⊂H0,ω,m2(I){S}_{h}\left(m)\subset {H}_{0,\omega ,m}^{2}\left(I), then from (4.6) and (4.9) we obtain λml≤λmhl{\lambda }_{m}^{l}\le {\lambda }_{mh}^{l}. Let Qh2,mW1,l{Q}_{h}^{2,m}{W}_{1,l}be the space spanned by Qh2,mum1,Qh2,mum2,…,Qh2,muml{Q}_{h}^{2,m}{u}_{m}^{1},{Q}_{h}^{2,m}{u}_{m}^{2},\ldots ,{Q}_{h}^{2,m}{u}_{m}^{l}. From the statements of Lemma 4.1 in [17], we know that Qh2,mW1,l{Q}_{h}^{2,m}{W}_{1,l}is an ll-dimensional subspace of Sh(m){S}_{h}\left(m). We derive from the minimax principle that λmhl≤maxvm∈Qh2,mW1,lAm(vm,vm)Bm(vm,vm)=maxvm∈W1,lAm(Qh2,mvm,Qh2,mvm)Bm(Qh2,mvm,Qh2,mvm).{\lambda }_{mh}^{l}\le \mathop{\max }\limits_{{v}_{m}\in {Q}_{h}^{2,m}{W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}=\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.From the bilinear property of Am(vm,vm){A}_{m}\left({v}_{m},{v}_{m}), we have Am(vm,vm)=Am(Qh2,mvm,Qh2,mvm)+2Am(vm−Qh2,mvm,Qh2,mvm)+Am(vm−Qh2,mvm,vm−Qh2,mvm){A}_{m}\left({v}_{m},{v}_{m})={A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})+2{A}_{m}({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})+{A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{v}_{m}-{Q}_{h}^{2,m}{v}_{m}). Further from Am(vm−Qh2,mvm,Qh2,mvm)=0{A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})=0and the non-negativity of Am(vm−Qh2,mvm,vm−Qh2,mvm){A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{v}_{m}-{Q}_{h}^{2,m}{v}_{m}), we have Am(Qh2,mvm,Qh2,mvm)≤Am(vm,vm).{A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})\le {A}_{m}\left({v}_{m},{v}_{m}).Thus, we obtain that λmhl≤maxvm∈W1,lAm(vm,vm)Bm(Qh2,mvm,Qh2,mvm)=maxvm∈W1,lAm(vm,vm)Bm(vm,vm)Bm(vm,vm)Bm(Qh2,mvm,Qh2,mvm)≤λmlmaxvm∈W1,lBm(vm,vm)Bm(Qh2,mvm,Qh2,mvm).\begin{array}{rcl}{\lambda }_{mh}^{l}& \le & \mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}\\ & =& \mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}\\ & \le & {\lambda }_{m}^{l}\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.\end{array}The proof is complete.□Define the interpolation operator Imh:H0,ω,m2(I)→Uh{I}_{mh}:{H}_{0,\omega ,m}^{2}\left(I)\to {U}_{h}by Imhum(t)=H3,m,i(t),t∈Ii,{I}_{mh}{u}_{m}\left(t)={H}_{3,m,i}\left(t),\hspace{1em}t\in {I}_{i},where Ii=[ti−1,ti]{I}_{i}=\left[{t}_{i-1},{t}_{i}], H3,mi(t){H}_{3,mi}\left(t)is a cubic Hermite interpolation polynomial of um{u}_{m}in Ii{I}_{i}. Let umi(t)=um(t),t∈Ii.{u}_{mi}\left(t)={u}_{m}\left(t),\hspace{1em}t\in {I}_{i}.From the remainder theorem of cubic Hermite interpolation, we have umi(t)−H3,m,i(t)=(umi)(4)(ξmi(t))4!(t−ti−1)2(t−ti)2,{u}_{mi}\left(t)-{H}_{3,m,i}\left(t)=\frac{{\left({u}_{mi})}^{\left(4)}\left({\xi }_{mi}\left(t))}{4\&#x0021;}{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2},where ξmi(t)∈Ii{\xi }_{mi}\left(t)\in {I}_{i}is a function depending on tt.Theorem 3Let Emi(t)=(umi)(4)(ξmi(t))4!{E}_{mi}\left(t)=\frac{{\left({u}_{mi})}^{\left(4)}\left({\xi }_{mi}\left(t))}{4\&#x0021;}, um∈H0,ω,m2(I){u}_{m}\in {H}_{0,\omega ,m}^{2}\left(I). Assume that um{u}_{m}is sufficiently smooth such that ∣∂tkEmi(t)∣≤M(k=0,1,2)| {\partial }_{t}^{k}{E}_{mi}\left(t)| \le M\left(k=0,1,2), where M is a constant greater than zero. Then the following inequality holds: (4.11)‖∂t2(Imhum−um)‖≲h2,\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m})\Vert \lesssim {h}^{2},where h=max1≤i≤n{hi}h={\max }_{1\le i\le n}\left\{{h}_{i}\right\}, hi=ti−ti−1{h}_{i}={t}_{i}-{t}_{i-1}, ‖um‖=∫Ium2dt12\Vert {u}_{m}\Vert ={\left[{\int }_{I}{u}_{m}^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}.ProofSince umi(t)−H3,m,i(t)=Emi(t)(t−ti−1)2(t−ti)2,{u}_{mi}\left(t)-{H}_{3,m,i}\left(t)={E}_{mi}\left(t){\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2},then we have ∂t2(umi(t)−H3,m,i(t))=∂t2Emi(t)(t−ti−1)2(t−ti)2+2∂tEmi(t)∂t[(t−ti−1)2(t−ti)2]+Emi(t)∂t2[(t−ti−1)2(t−ti)2].{\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t))={\partial }_{t}^{2}{E}_{mi}\left(t){\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}+2{\partial }_{t}{E}_{mi}\left(t){\partial }_{t}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]+{E}_{mi}\left(t){\partial }_{t}^{2}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}].Thus, we obtain ∣∂t2(umi(t)−H3,m,i(t))∣2≲[(t−ti−1)2(t−ti)2]2+[∂t((t−ti−1)2(t−ti)2)]2+{∂t2[(t−ti−1)2(t−ti)2]}2=[(t−ti−1)(t−ti)]4+4[(t−ti−1)(t−ti)(2t−ti−1−ti)]2+4[4(t−ti−1)(t−ti)+(t−ti−1)2+(t−ti)2]2≤hi28+4hihi222+8hi22+hi22≲hi4.\begin{array}{rcl}| {\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t)){| }^{2}& \lesssim & {\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]}^{2}+{\left[{\partial }_{t}\left({\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2})]}^{2}+{\left\{{\partial }_{t}^{2}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]\right\}}^{2}\\ & =& {\left[\left(t-{t}_{i-1})\left(t-{t}_{i})]}^{4}+4{\left[\left(t-{t}_{i-1})\left(t-{t}_{i})\left(2t-{t}_{i-1}-{t}_{i})]}^{2}+4{\left[4\left(t-{t}_{i-1})\left(t-{t}_{i})+{\left(t-{t}_{i-1})}^{2}+{\left(t-{t}_{i})}^{2}]}^{2}\\ & \le & {\left(\frac{{h}_{i}}{2}\right)}^{8}+4{\left[{h}_{i}{\left(\frac{{h}_{i}}{2}\right)}^{2}\right]}^{2}+{\left[8{\left(\frac{{h}_{i}}{2}\right)}^{2}+{h}_{i}^{2}\right]}^{2}\lesssim {h}_{i}^{4}.\end{array}Thus, ‖∂t2(Imhum−um)‖2=∑i=1n∫Ii[∂t2(umi(t)−H3,m,i(t))]2dt≲∑i=1nhi5≲h4.\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m}){\Vert }^{2}=\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{I}_{i}}{\left[{\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t))]}^{2}{\rm{d}}t\lesssim \mathop{\sum }\limits_{i=1}^{n}{h}_{i}^{5}\lesssim {h}^{4}.Furthermore, we have ‖∂t2(Imhum−um)‖≲h2.\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m})\Vert \lesssim {h}^{2}.The proof is complete.□Theorem 4Let λmhl{\lambda }_{mh}^{l}be the approximate eigenvalue of λml{\lambda }_{m}^{l}. Assume that um∈H0,ω,m2(I){u}_{m}\in {H}_{0,\omega ,m}^{2}\left(I)and satisfies the condition of Theorem 3, then the following inequality holds: ∣λmhl−λml∣≲h4,| {\lambda }_{mh}^{l}-{\lambda }_{m}^{l}| \lesssim {h}^{4},where c(l)c\left(l)is a constant independent of h.ProofFor brief, we only give the proof for the case of m≠1m\ne 1, and it can be similarly proven for the case of m=1m=1. For ∀q∈W1,l\forall \hspace{-0.25em}q\in {W}_{1,l}, we have q=∑i=1lqiumiq={\sum }_{i=1}^{l}{q}_{i}{u}_{m}^{i}. By using the orthogonality of the characteristics function umi{u}_{m}^{i}and Bm(umi,umi)=1{B}_{m}\left({u}_{m}^{i},{u}_{m}^{i})=1, we have Bm(q,q)−Bm(Qh2,mq,Qh2,mq)Bm(q,q)≤2∣Bm(q,q−Qh2,mq)∣Bm(q,q)≤2∑i,j=1l∣qiqjBm(umi−Qh2,mumi,umj)∣∑i=1l∣qi∣2≤2lmaxi,j=1,…,l∣Bm(umi−Qh2,mumi,umj)∣.\begin{array}{rcl}\frac{{B}_{m}\left(q,q)-{B}_{m}\left({Q}_{h}^{2,m}q,{Q}_{h}^{2,m}q)}{{B}_{m}\left(q,q)}& \le & \frac{2| {B}_{m}\left(q,q-{Q}_{h}^{2,m}q)| }{{B}_{m}\left(q,q)}\\ & \le & \frac{2{\displaystyle \sum }_{i,j=1}^{l}| {q}_{i}{q}_{j}{B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| }{{\displaystyle \sum }_{i=1}^{l}| {q}_{i}{| }^{2}}\\ & \le & 2l\mathop{\max }\limits_{i,j=1,\ldots ,l}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| .\end{array}From Cauchy-Schwarz inequality we have ∣Bm(umi−Qh2,mumi,umj)∣=1λmj∣λmjBm(umj,umi−Qh2,mumi)∣=1λmj∣Am(umj,umi−Qh2,mumi)∣=1λmj∣Am(umj−Qh2,mumj,umi−Qh2,mumi)∣≤1λmj[Am(umj−Qh2,mumj,umj−Qh2,mumj)]12[Am(umi−Qh2,mumi,umi−Qh2,mumi)]12.\begin{array}{rcl}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| & =& \frac{1}{{\lambda }_{m}^{j}}| {\lambda }_{m}^{j}{B}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & =& \frac{1}{{\lambda }_{m}^{j}}| {A}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & =& \frac{1}{{\lambda }_{m}^{j}}| {A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j})]}^{\tfrac{1}{2}}{\left[{A}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})]}^{\tfrac{1}{2}}.\end{array}When m≥2m\ge 2, from Hardy inequality (cf. B8.6 in [16]) we derive ∫I1(t+1)3(umi)2dt≲∫I1t+1(∂tumi)2dt,\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\left({u}_{m}^{i})}^{2}{\rm{d}}t\lesssim \mathop{\int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t,\hspace{3.2em}∫I1(t+1)2(∂tumi)2dt≲∫I(∂t2umi)2dt.\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\lesssim \mathop{\int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t.\hspace{4.7em}Since [∂tumi(1)]2=14∫I∂t((t+1)∂tumi)dt2=14∫I∂tumi+(t+1)∂t2umidt2≲∫I(∂tumi)2dt+∫I(t+1)(∂t2umi)2dt≲∫I1t+1(∂tumi)2dt+∫I(t+1)(∂t2umi)2dt,\begin{array}{rcl}{\left[{\partial }_{t}{u}_{m}^{i}\left(1)]}^{2}& =& \frac{1}{4}{\left[\mathop{\displaystyle \int }\limits_{I}{\partial }_{t}\left(\left(t+1){\partial }_{t}{u}_{m}^{i}){\rm{d}}t\right]}^{2}\\ & =& \frac{1}{4}{\left[\mathop{\displaystyle \int }\limits_{I}{\partial }_{t}{u}_{m}^{i}+\left(t+1){\partial }_{t}^{2}{u}_{m}^{i}{\rm{d}}t\right]}^{2}\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t,\end{array}\hspace{8.15em}from Lemma 1 we have ‖umi‖2,ω,m2≲∫I(t+1)(∂t2umi)2dt+∫I1t+1(∂tumi)2dt+∫I1(t+1)3(umi)2dt≲∫I(∂t2umi)2dt+∫I1t+1(∂tumi)2dt≲∫I(∂t2umi)2dt+∫I1(t+1)2(∂tumi)2dt≲∫I(∂t2umi)2dt.\begin{array}{rcl}\Vert {u}_{m}^{i}{\Vert }_{2,\omega ,m}^{2}& \lesssim & \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\left({u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t.\end{array}Then we derive that ∣Bm(umi−Qh2,mumi,umj)∣=1λmj∣λmjBm(umj,umi−Qh2,mumi)∣≤1λmj[Am(umj−Qh2,mumj,umj−Qh2,mumj)]12[Am(umi−Qh2,mumi,umi−Qh2,mumi)]12≤1λmj[Am(umj−Imhumj,umj−Imhumj)]12⋅[Am(umi−Imhumi,umi−Imhumi)]12≤Mλmj‖umj−Ihumj‖2,ω,m‖umi−Ihumi‖2,ω,m≲∫I[∂t2(umj−Imhumj)]2dt12∫I[∂t2(umi−Imhumi)]2dt12≲‖∂t2(umj−Imhumj)‖⋅‖∂t2(umi−Imhumi)‖.\begin{array}{rcl}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| & =& \frac{1}{{\lambda }_{m}^{j}}| {\lambda }_{m}^{j}{B}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j})]}^{\tfrac{1}{2}}{\left[{A}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})]}^{\tfrac{1}{2}}\\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j},{u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})]}^{\tfrac{1}{2}}\cdot {\left[{A}_{m}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i},{u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})]}^{\tfrac{1}{2}}\\ & \le & \frac{M}{{\lambda }_{m}^{j}}\Vert {u}_{m}^{j}-{I}_{h}{u}_{m}^{j}{\Vert }_{2,\omega ,m}\Vert {u}_{m}^{i}-{I}_{h}{u}_{m}^{i}{\Vert }_{2,\omega ,m}\\ & \lesssim & {\left(\mathop{\displaystyle \int }\limits_{I}{\left[{\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})]}^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}{\left(\mathop{\displaystyle \int }\limits_{I}{\left[{\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})]}^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}\\ & \lesssim & \Vert {\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})\Vert \cdot \Vert {\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})\Vert .\end{array}Similarly, when m=0m=0, we derive that ∣Bm(umi−Qh2,mumi,umj)∣≲‖∂t2(umj−Imhumj)‖⋅‖∂t2(umi−Imhumi)‖.| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| \lesssim \Vert {\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})\Vert \cdot \Vert {\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})\Vert .Since Bm(q,q)Bm(Qh2,mq,Qh2,mq)≤11−2lmaxi,j=1,…,l∣Bm(umi−Qh2,mumi,umj)∣,\frac{{B}_{m}\left(q,q)}{{B}_{m}\left({Q}_{h}^{2,m}q,{Q}_{h}^{2,m}q)}\le \frac{1}{1-2l{\max }_{i,j=1,\ldots ,l}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| },we obtain from Theorems 2 and 3 the desired results.□5Efficient implementation of the algorithmIn order to efficiently solve the problems (3.2), we start by constructing a set of basis functions which satisfy boundary conditions. Let φ00(t)=2t−t0h1+1t−t0h1−12,ti≤t≤ti+1,0,others,φ01(t)=h1−2(t−t0)(t−t1)2,t0≤t≤t10,others,φi0(t)=1+t−tihi21−2t−tihi,ti−1≤t≤ti,2t−tihi+1+1t−tihi+1−12,ti≤t≤ti+1,0,others,φi1(t)=hi−2(t−ti)(t−ti−1)2,ti−1≤t≤ti,hi+1−2(t−ti)(t−ti+1)2,ti≤t≤ti+1,0,others,φn1(t)=hn−2(t−tn)(t−tn−1)2,tn−1≤t≤tn,0,others,\begin{array}{rcl}{\varphi }_{0}^{0}\left(t)& =& \left\{\begin{array}{ll}\left(2\frac{t-{t}_{0}}{{h}_{1}}+1\right){\left(\frac{t-{t}_{0}}{{h}_{1}}-1\right)}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{0}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{1}^{-2}\left(t-{t}_{0}){\left(t-{t}_{1})}^{2},& {t}_{0}\le t\le {t}_{1}\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{i}^{0}\left(t)& =& \left\{\begin{array}{ll}{\left(1+\frac{t-{t}_{i}}{{h}_{i}}\right)}^{2}\left(1-2\frac{t-{t}_{i}}{{h}_{i}}\right),& {t}_{i-1}\le t\le {t}_{i},\\ \left(2\frac{t-{t}_{i}}{{h}_{i+1}}+1\right){\left(\frac{t-{t}_{i}}{{h}_{i+1}}-1\right)}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{i}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{i}^{-2}\left(t-{t}_{i}){\left(t-{t}_{i-1})}^{2},& {t}_{i-1}\le t\le {t}_{i},\\ {h}_{i+1}^{-2}\left(t-{t}_{i}){\left(t-{t}_{i+1})}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{n}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{n}^{-2}\left(t-{t}_{n}){\left(t-{t}_{n-1})}^{2},& {t}_{n-1}\le t\le {t}_{n},\\ 0,& {\rm{others}},\end{array}\right.\end{array}where i=1,…,n−1i=1,\ldots ,n-1. It is clear that Sh(0)=span{φ00(t),…,φn−10(t),φ11(t),…,φn1(t)};Sh(1)=span{φ10(t),…,φn−10(t),φ01(x),…,φn1(t)};Sh(m)=span{φ10(t),…,φn−10(t),φ11(x),…,φn1(t)},(m≥2).\begin{array}{l}{S}_{h}\left(0)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{0}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{1}^{1}\left(t),\ldots ,{\varphi }_{n}^{1}\left(t)\right\};\\ {S}_{h}\left(1)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{1}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{0}^{1}\left(x),\ldots ,{\varphi }_{n}^{1}\left(t)\right\};\\ {S}_{h}\left(m)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{1}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{1}^{1}\left(x),\ldots ,{\varphi }_{n}^{1}\left(t)\right\},\left(m\ge 2).\end{array}Denote aijpq=∫I(t+1)(φjp)″(φiq)″dt,bijpq=∫I1t+1(φjp)′(φiq)′dt,cijpq=∫I1(t+1)3φjpφiqdt,dijpq=φjp(1)φiq(1),\begin{array}{rcl}{a}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\varphi }_{j}^{p})}^{^{\prime\prime} }{\left({\varphi }_{i}^{q})}^{^{\prime\prime} }{\rm{d}}t,\hspace{1em}{b}_{ij}^{pq}=\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\varphi }_{j}^{p})}^{^{\prime} }{\left({\varphi }_{i}^{q})}^{^{\prime} }{\rm{d}}t,\\ {c}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\varphi }_{j}^{p}{\varphi }_{i}^{q}{\rm{d}}t,\hspace{1em}{d}_{ij}^{pq}={\varphi }_{j}^{p}\left(1){\varphi }_{i}^{q}\left(1),\end{array}where p,q=0,1p,q=0,1.Next, we will derive the matrix form of the discrete variational scheme (3.2).Case 1. When m=0m=0, let (5.1)u0h=u00φ00+un1φn1+∑i=1n−1(ui0φi0+ui1φi1).{u}_{0h}={u}_{0}^{0}{\varphi }_{0}^{0}+{u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).Plugging the expression (5.1) in (3.2) and taking v0h{v}_{0h}through all the basis functions in Sh(0){S}_{h}\left(0), we derive that (5.2)(A0+B0+D0)U0=λ0hRD0U0,\left({A}_{0}+{B}_{0}+{D}_{0}){U}^{0}={\lambda }_{0h}R{D}_{0}{U}^{0},where A0=(aij00)(aij10)(aij01)(aij11),B0=(bij00)(bij10)(bij01)(bij11),D0=(dij00)(dij10)(dij01)(dij11),{A}_{0}=\left(\begin{array}{cc}\left({a}_{ij}^{00})& \left({a}_{ij}^{10})\\ \left({a}_{ij}^{01})& \left({a}_{ij}^{11})\end{array}\right),\hspace{1.0em}{B}_{0}=\left(\begin{array}{cc}\left({b}_{ij}^{00})& \left({b}_{ij}^{10})\\ \left({b}_{ij}^{01})& \left({b}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{0}=\left(\begin{array}{cc}\left({d}_{ij}^{00})& \left({d}_{ij}^{10})\\ \left({d}_{ij}^{01})& \left({d}_{ij}^{11})\end{array}\right),\hspace{1.0em}U0=(u00,…,uN−10,u11,…,un1)T.{U}^{0}={\left({u}_{0}^{0},\ldots ,{u}_{N-1}^{0},{u}_{1}^{1},\ldots ,{u}_{n}^{1})}^{T}.\hspace{0.4em}Similarly, when m=1m=1, let (5.3)u1h=u01φ01+un1φn1+∑i=1n−1(ui0φi0+ui1φi1).{u}_{1h}={u}_{0}^{1}{\varphi }_{0}^{1}+{u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).\hspace{0.5em}Plugging the expression (5.3) in (3.2) and taking v1h{v}_{1h}through all the basis functions in Sh(1){S}_{h}\left(1), we obtain (5.4)[A1+(1+m)D1]U1=λ1hRD1U1,\left[{A}_{1}+\left(1+m){D}_{1}]{U}^{1}={\lambda }_{1h}R{D}_{1}{U}^{1},where A1=(a˜ij00)(a˜ij10)(a˜ij01)(a˜ij11),D1=(d˜ij00)(d˜ij10)(d˜ij01)(d˜ij11),\hspace{4em}{A}_{1}=\left(\begin{array}{cc}\left({\tilde{a}}_{ij}^{00})& \left({\tilde{a}}_{ij}^{10})\\ \left({\tilde{a}}_{ij}^{01})& \left({\tilde{a}}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{1}=\left(\begin{array}{cc}\left({\tilde{d}}_{ij}^{00})& \left({\tilde{d}}_{ij}^{10})\\ \left({\tilde{d}}_{ij}^{01})& \left({\tilde{d}}_{ij}^{11})\end{array}\right),\hspace{2.55em}a˜ijpq=∫I(t+1)L1φjpL1φiqdt,d˜ijpq=φjp(1)φiq(1),U1=(u10…,un−10,u01,…,un1)T.\begin{array}{rcl}{\tilde{a}}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{1}{\varphi }_{j}^{p}{L}_{1}{\varphi }_{i}^{q}{\rm{d}}t,\hspace{1em}{\tilde{d}}_{ij}^{pq}={\varphi }_{j}^{p}\left(1){\varphi }_{i}^{q}\left(1),\\ {U}^{1}& =& {\left({u}_{1}^{0}\ldots ,{u}_{n-1}^{0},{u}_{0}^{1},\ldots ,{u}_{n}^{1})}^{T}.\end{array}When m≥2m\ge 2, let (5.5)u2h=un1φn1+∑i=1n−1(ui0φi0+ui1φi1).\hspace{3.1em}{u}_{2h}={u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).\hspace{2.95em}Plugging the expression (5.5) in (3.2) and taking vmh{v}_{mh}through all the basis functions in Sh(m),{S}_{h}\left(m),we gain (5.6)[Am+(2m2+1)Bm+m2(m2−4)Cm+Dm]Um=λmhRDmUm,\left[{A}_{m}+\left(2{m}^{2}+1){B}_{m}+{m}^{2}\left({m}^{2}-4){C}_{m}+{D}_{m}]{U}^{m}={\lambda }_{mh}R{D}_{m}{U}^{m},where (5.7)Am=(aij00)(aij10)(aij01)(aij11),Bm=(bij00)(bij10)(bij01)(bij11),Cm=(cij00)(cij10)(cij01)(cij11),Dm=(dij00)(dij10)(dij01)(dij11){A}_{m}=\left(\begin{array}{cc}\left({a}_{ij}^{00})& \left({a}_{ij}^{10})\\ \left({a}_{ij}^{01})& \left({a}_{ij}^{11})\end{array}\right),\hspace{1.0em}{B}_{m}=\left(\begin{array}{cc}\left({b}_{ij}^{00})& \left({b}_{ij}^{10})\\ \left({b}_{ij}^{01})& \left({b}_{ij}^{11})\end{array}\right),\hspace{1.0em}\hspace{1em}{C}_{m}=\left(\begin{array}{cc}\left({c}_{ij}^{00})& \left({c}_{ij}^{10})\\ \left({c}_{ij}^{01})& \left({c}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{m}=\left(\begin{array}{cc}\left({d}_{ij}^{00})& \left({d}_{ij}^{10})\\ \left({d}_{ij}^{01})& \left({d}_{ij}^{11})\end{array}\right)Um=(u10,…,un−10,u11,…,un1)T.{U}^{m}={\left({u}_{1}^{0},\ldots ,{u}_{n-1}^{0},{u}_{1}^{1},\ldots ,{u}_{n}^{1})}^{T}.Note that we know from the properties of cubic hermit interpolation basis function that the stiff matrices and mass matrices in (5.4)–(5.6) are all sparse. Thus, they can be efficiently solved.6Numerical experimentsIn order to show the accuracy and convergence of the proposed algorithm, we will carry out a series of numerical tests. We operate our programs in MATLAB 2016b.Example 1We take R=1R=1and m=0,1,2,3m=0,1,2,3. The eigenvalues for different mmand hhare listed in Table 1.Table 1Eigenvalues for m=0,1,2,3m=0,1,2,3and different hhhλ0h{\lambda }_{0h}λ1h{\lambda }_{1h}λ2h{\lambda }_{2h}λ3h{\lambda }_{3h}1/82.0000000244610734.0000000730055186.0000252085831298.0003070914540561/162.0000000158994934.0000000724494046.0000015395142628.0000191017624241/322.0000000064929494.0000000849451286.0000000956013148.0000011925251611/642.0000000026608134.0000000910470356.0000000059450748.000000074519297We know from Table 1 that the eigenvalues achieve at least six-digit accuracy with h≤132h\le \frac{1}{32}for m=0,1,2,3m=0,1,2,3. In order to further show the convergence of the algorithm, we choose the numerical solutions of h=164h=\frac{1}{64}as reference solutions, and the error figures of the approximate eigenvalues λmh(m=0,1,2,3){\lambda }_{mh}\left(m=0,1,2,3)with different hhare presented in Figure 1. We observe from Figure 1 that the numerical eigenvalues are also convergent.Figure 1Errors between numerical solutions and the reference solution for R=1R=1.Example 2We take R=2R=2and m=0,1,2,3m=0,1,2,3. The eigenvalues for different mmand hhare listed in Table 2.Table 2Eigenvalues for m=0,1,2,3m=0,1,2,3and different hhhλ0h{\lambda }_{0h}λ1h{\lambda }_{1h}λ2h{\lambda }_{2h}λ3h{\lambda }_{3h}1/81.0000000122305372.0000000365027593.0000126042915654.0001535457270281/161.0000000079497472.0000000362247023.0000007697571314.0000095508812121/321.0000000032464742.0000000424725643.0000000478006574.0000005962625811/641.0000000013304062.0000000455235173.0000000029725374.000000037259649Similarly, we observe from Table 2 that the eigenvalues have at least six-digit accuracy with h≤132h\le \frac{1}{32}for m=0,1,2,3m=0,1,2,3. We still choose the numerical solutions of h=164h=\frac{1}{64}as reference solutions, the error figures of the approximate eigenvalues λmh(m=0,1,2,3){\lambda }_{mh}\left(m=0,1,2,3)with different hhare listed in Figure 2. We see from Figure 2 that the approximation eigenvalues are also convergent. Besides, in order to show the convergence rate of our algorithm more intuitively, we also plot the error figures in semilog scale in Figures 3 and 4.Figure 2Errors between numerical solutions and the reference solution for R=2R=2.Figure 3Error curves in semilog scale between the numerical solution and the reference solution for R=1R=1.Figure 4Error curves in semilog scale between the numerical solution and the reference solution for R=2R=2.Next, we shall provide a numerical example for some larger Fourier norm mm.Example 3We take R=1R=1and m=4,5,6,7m=4,5,6,7. The eigenvalues for different mmand hhare listed in Table 3.Table 3Eigenvalues for m=4,5,6,7m=4,5,6,7and different hhhhλ4h{\lambda }_{4h}λ5h{\lambda }_{5h}λ6h{\lambda }_{6h}λ7h{\lambda }_{7h}1/410.023827239467012.075633841422614.187693723539216.39313166639011/810.001490600395312.004878672365214.012611174554816.02775272290221/1610.000093110825312.000307476583914.000803468087416.00179279509841/3210.000005818033112.000019258364014.000050462262416.00011299986481/6410.000000363585112.000001204285914.000003157754116.00000707750631/12810.000000022918412.000000075660014.000000197488516.0000004427145Likewise, we observe from Table 3 that the eigenvalues have at least five-digit accuracy with h≤164h\le \frac{1}{64}for m=4,5,6,7m=4,5,6,7. We choose the numerical solutions of h=1128h=\frac{1}{128}as reference solutions, and the error figures of the approximate eigenvalues λmh(m=4,5,6,7){\lambda }_{mh}\left(m=4,5,6,7)with different hhare listed in Figure 5. We see from Figure 5 that the approximation eigenvalues are also convergent.Figure 5Errors between numerical solutions and the reference solution for m=4,5,6,7m=4,5,6,7and R=1R=1.Introducing the usual definition of convergence order: (6.1)c(h)=log2∣λh−λh2∣∣λh2−λh4∣.c\left(h)={\log }_{2}\left(\frac{| {\lambda }_{h}-{\lambda }_{\tfrac{h}{2}}| }{| {\lambda }_{\tfrac{h}{2}}-{\lambda }_{\tfrac{h}{4}}| }\right).For brevity, we shall use formula (6.1) to calculate the convergence order of the approximation eigenvalues of m=4,5,6,7m=4,5,6,7in the unit disk and list them in Table 4. We observe from Table 4 that the convergence order is about 4.Table 4Convergence order c(h)c\left(h)for the eigenvalues with m=4,5,6,7m=4,5,6,7hhλ4h{\lambda }_{4h}λ5h{\lambda }_{5h}λ6h{\lambda }_{6h}λ7h{\lambda }_{7h}1/23.99243.84593.69603.57701/43.99853.95223.89023.81501/84.00083.98733.97093.94991/164.00033.99683.99263.98721/324.00103.99973.99823.9968Finally, we plot the eigenvalue error in log-log scale to describe the algebra convergence rate in Figure 6.Figure 6The eigenvalue errors in log-log scale between numerical solutions and the reference solution for m=4,5,6,7m=4,5,6,7and R=1R=1.7ConclusionWe present in this article a novel finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. The main advantage of this method is that the original problem is transformed into a series of one-dimensional problems which can be solved in parallel. Then, by introducing the polar conditions and the weighted Sobolev space, we prove the error estimates of approximation eigenvalues by using the minimax principle. The method developed in this article can be applied to more complex problems or more general polar geometric domains which will be the subject of our future endeavors. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem

Zhang, Hui; Liu, Zixin; Zhang, Jun
Open Mathematics , Volume 20 (1): 16 – Jan 1, 2022
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de Gruyter
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© 2022 Hui Zhang et al., published by De Gruyter
ISSN
2391-5455
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2391-5455
DOI
10.1515/math-2022-0032
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Abstract

1IntroductionFourth-order Steklov eigenvalue problems with eigenvalue parameter in boundary conditions are widely used in mathematics and physics, such as the surface wave research, the stability analysis of mechanical oscillator in a viscous fluid, the study of vibration mode of structure in contact with an incompressible fluid, and so on [1,2,3, 4,5]. The first eigenvalue λ1{\lambda }_{1}also plays an important role in the positivity-preserving properties for the biharmonic-operator Δ2{\Delta }^{2}under the boundary conditions ϕ∣∂Ω=(Δϕ−λϕν)∣∂Ω=0\phi \hspace{-0.25em}{| }_{\partial \Omega }=\left(\Delta \phi -\lambda {\phi }_{\nu }){| }_{\partial \Omega }=0in [6,7].There are many existing results about the fourth-order Steklov eigenvalue problems, but they mainly focus on the qualitative analysis. Kuttler [8] proved that the first eigenvalue is simple and the corresponding eigenfunction does not change the sign. Ferrero et al. [7] and Bucur et al. [9] studied the spectrum on a bounded domain, and the explicit representation of the spectrum is given when the domain is a ball. Recently, the existence of an optimal convex shape among domains of a given measure is proved in [10], and the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues also is established in [11]. For the numerical methods of the fourth-order Steklov eigenvalue problems, a conforming finite element method was first proposed in [12], then some spectral methods are also developed [13].As we all know, if the conforming finite element method is directly used to solve a fourth-order problem, the boundary of the element requires the continuity of the first derivative, which not only brings the difficulty of constructing the basis function but also costs a lot of calculation time and memory capacity, especially for some special regions, such as circular region, spherical region, and so on. How to efficiently solve a fourth-order Steklov eigenvalue problem in a circular domain? To the best of our knowledge, there are few reports on using some efficient numerical to solve this problem. Thus, the aim of this article is to propose an effective finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.This article is organized as follows. In Section 2, a reduced scheme based on polar coordinate transformation is presented. In Section 3, the weighted space and discrete variational form are derived. In Section 4, the error estimation of approximation solutions is proved. In Section 5, we present the process of effective implementation of the algorithm. We present some numerical experiments in Section 6 to illustrate the accuracy and efficiency of our proposed algorithm. Finally, we give in Section 7 some concluding remarks.2Reduced scheme based on polar coordinate transformationThe fourth-order Steklov eigenvalue problems read: (2.1)Δ2ϕ=0,inΩ,{\Delta }^{2}\phi =0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,(2.2)ϕ=0,on∂Ω,\phi =0,\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,(2.3)Δϕ=λ∂ϕ∂ν,on∂Ω,\Delta \phi =\lambda \frac{\partial \phi }{\partial \nu },\hspace{1em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,where Ω={(x,y)∈R2:0<x2+y2<R}\Omega =\left\{\left(x,y)\in {{\mathbb{R}}}^{2}:0\lt \sqrt{{x}^{2}+{y}^{2}}\lt R\right\}, ν\nu is the unit outward normal to the boundary ∂Ω\partial \Omega . Let x=rcosθ,y=rsinθx=r\cos \theta ,y=r\sin \theta , ψ(r,θ)=ϕ(x,y)\psi \left(r,\theta )=\phi \left(x,y). Then we derive that (2.4)Δϕ(x,y)=Lψ(r,θ)=1r∂∂rr∂ψ(r,θ)∂r+1r2∂2ψ(r,θ)∂θ2.\Delta \phi \left(x,y)=L\psi \left(r,\theta )=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial \psi \left(r,\theta )}{\partial r}\right)+\frac{1}{{r}^{2}}\frac{{\partial }^{2}\psi \left(r,\theta )}{\partial {\theta }^{2}}.Then the equivalent form of (2.1)–(2.3) in polar coordinates is as follows: (2.5)L2ψ(r,θ)=0,inD=(0,R)×[0,2π),{L}^{2}\psi \left(r,\theta )=0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{\mathcal{D}}=\left(0,R)\times \left[0,2\pi ),(2.6)ψ(R,θ)=0,θ∈[0,2π),\psi \left(R,\theta )=0,\hspace{1em}\theta \in \left[0,2\pi ),(2.7)Lψ(R,θ)=λ∂ψ∂r(R,θ),θ∈[0,2π).L\psi \left(R,\theta )=\lambda \frac{\partial \psi }{\partial r}\left(R,\theta ),\hspace{1em}\theta \in \left[0,2\pi ).\hspace{1.75em}Since ψ(r,θ)\psi \left(r,\theta )is 2π2\pi periodic in θ\theta , then we have (2.8)ψ(r,θ)=∑∣m∣=0∞ψm(r)eimθ.\psi \left(r,\theta )=\mathop{\sum }\limits_{| m| =0}^{\infty }{\psi }_{m}\left(r){{\rm{e}}}^{im\theta }.Substituting (2.8) into (2.4), we derive that (2.9)Lψ(r,θ)=∑∣m∣=0∞1r∂∂rr∂ψm(r)∂r−m2rψm(r)eimθ.L\psi \left(r,\theta )=\mathop{\sum }\limits_{| m| =0}^{\infty }\frac{1}{r}\left\{\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]-\frac{{m}^{2}}{r}{\psi }_{m}\left(r)\right\}{{\rm{e}}}^{im\theta }.Following the discussion in [14,15], to overcome the pole singularity introduced by polar coordinate transformation, we need to introduce the essential pole conditions, which make (2.9) meaningful, as follows: (2.10)m2ψm(0)=0,limr→0+∂∂rr∂ψm(r)∂r−m2rψm(r)=0.{m}^{2}{\psi }_{m}\left(0)=0,\mathop{\mathrm{lim}}\limits_{r\to {0}^{+}}\left\{\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]-\frac{{m}^{2}}{r}{\psi }_{m}\left(r)\right\}=0.Using the fact that ∂∂rr∂ψm(r)∂r=∂ψm(r)∂r+r∂2ψm(r)∂r2\frac{\partial }{\partial r}\left[r\frac{\partial {\psi }_{m}\left(r)}{\partial r}\right]=\frac{\partial {\psi }_{m}\left(r)}{\partial r}+r\frac{{\partial }^{2}{\psi }_{m}\left(r)}{\partial {r}^{2}}, (2.10) can be reduced to (2.11)m2ψm(0)=0,(1−m2)∂ψm(0)∂r=0.{m}^{2}{\psi }_{m}\left(0)=0,\hspace{1em}\left(1-{m}^{2})\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0.From (2.11) we can further obtain that (2.12)(1)∂ψm(0)∂r=0,(m=0);\left(1)\hspace{1em}\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0,\hspace{1em}\left(m=0);\hspace{6.25em}(2.13)(2)ψm(0)=0,(∣m∣=1);\left(2)\hspace{1em}{\psi }_{m}\left(0)=0,\hspace{1em}\left(| m| =1);\hspace{6.65em}(2.14)(3)ψm(0)=0,∂ψm(0)∂r=0,(∣m∣≥2).\left(3)\hspace{1em}{\psi }_{m}\left(0)=0,\hspace{1em}\frac{\partial {\psi }_{m}\left(0)}{\partial r}=0,\hspace{1em}\left(| m| \ge 2).Let r=t+12R,um(t)=ψm(r),Lmum(t)=1t+1∂∂tt∂um(t)∂t−m2(t+1)2um(t)r=\frac{t+1}{2}R,{u}_{m}\left(t)={\psi }_{m}\left(r),{L}_{m}{u}_{m}\left(t)=\frac{1}{t+1}\frac{\partial }{\partial t}\left[t\frac{\partial {u}_{m}\left(t)}{\partial t}\right]-\frac{{m}^{2}}{{\left(t+1)}^{2}}{u}_{m}\left(t). From (2.8), (2.11)–(2.14), and the orthogonal properties of Fourier basis functions, (2.5)–(2.7) are equivalent to a series of one-dimensional eigenvalue problems (2.15)Lm2um(t)=0,t∈(−1,1),{L}_{m}^{2}{u}_{m}\left(t)=0,\hspace{1em}t\in \left(-1,1),(2.16)(1)∂um(−1)∂t=0,um(1)=0,Lmum(1)=R2λm∂um(1)∂t,(m=0);\left(1)\hspace{1em}\frac{\partial {u}_{m}\left(-1)}{\partial t}=0,\hspace{1em}{u}_{m}\left(1)=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(m=0);\hspace{.1em}(2.17)(2)um(−1)=0,um(1)=0,Lmum(1)=R2λm∂um(1)∂t,(∣m∣=1);\left(2)\hspace{1em}{u}_{m}\left(-1)=0,\hspace{1em}{u}_{m}\left(1)=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(| m| =1);\hspace{.5em}(2.18)(3)um(±1)=0,∂um(−1)∂t=0,Lmum(1)=R2λm∂um(1)∂t,(∣m∣≥2).\left(3)\hspace{1em}{u}_{m}\left(\pm 1)=0,\hspace{1em}\frac{\partial {u}_{m}\left(-1)}{\partial t}=0,\hspace{1em}{L}_{m}{u}_{m}\left(1)=\frac{R}{2}{\lambda }_{m}\frac{\partial {u}_{m}\left(1)}{\partial t},\hspace{1em}\left(| m| \ge 2).3Weighted space and discrete variational formWithout losing generality, we only consider the case of m≥0m\ge 0. First, we divide the solution interval I=(−1,1)I=\left(-1,1)as follows: −1=t0<t1<⋯<ti<⋯<tn=1.-1={t}_{0}\lt {t}_{1}\hspace{0.33em}\lt \cdots \lt {t}_{i}\hspace{0.33em}\lt \cdots \lt {t}_{n}=1.Define the usual weighted Sobolev space: Lω2(I)≔ρ:∫Iωρ2dt<∞{L}_{\omega }^{2}\left(I):= \left\{\rho :\mathop{\int }\limits_{I}\omega {\rho }^{2}{\rm{d}}t\lt \infty \right\}equipped with the following inner product and norm: (ρ,v)ω=∫Iωρvdt,‖ρ‖w=∫Iωρ2dt12,{\left(\rho ,v)}_{\omega }=\mathop{\int }\limits_{I}\omega \rho v{\rm{d}}t,\hspace{1em}\Vert \rho {\Vert }_{w}={\left(\mathop{\int }\limits_{I}\omega {\rho }^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}},where ω=1+t,t∈(−1,1)\omega =1+t,t\in \left(-1,1). We further introduce the following weighted Sobolev space: H0,ω,m2(I)≔um:Lmum∈Lω2(I),m2um(−1)=(1−m2)∂um(−1)∂t=um(1)=0,{H}_{0,\omega ,m}^{2}\left(I):= \left\{{u}_{m}:{L}_{m}{u}_{m}\in {L}_{\omega }^{2}\left(I),{m}^{2}{u}_{m}\left(-1)=\left(1-{m}^{2})\frac{\partial {u}_{m}\left(-1)}{\partial t}={u}_{m}\left(1)=0\right\},equipped with the inner product and norm: (um,vm)2,ω,m=(Lmum,Lmvm)ω,‖um‖2,ω,m=(um,um)2,ω,m.{\left({u}_{m},{v}_{m})}_{2,\omega ,m}={\left({L}_{m}{u}_{m},{L}_{m}{v}_{m})}_{\omega },\Vert {u}_{m}{\Vert }_{2,\omega ,m}=\sqrt{{\left({u}_{m},{u}_{m})}_{2,\omega ,m}}.Then the variational form of (2.15)–(2.18) is: Find (λm,um≠0)∈R×H0,ω,m2(I)\left({\lambda }_{m},{u}_{m}\ne 0)\in {\mathbb{R}}\times {H}_{0,\omega ,m}^{2}\left(I), such that (3.1)Am(um,vm)=λmBm(um,vm),∀vm∈H0,ω,m2(I),{A}_{m}\left({u}_{m},{v}_{m})={\lambda }_{m}{B}_{m}\left({u}_{m},{v}_{m}),\hspace{1em}\forall {v}_{m}\in {H}_{0,\omega ,m}^{2}\left(I),where Am(um,vm)=∫I(t+1)LmumLmvmdt,Bm(um,vm)=R∂um(1)∂t∂vm(1)∂t.\begin{array}{rcl}{A}_{m}\left({u}_{m},{v}_{m})& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t,\\ {B}_{m}\left({u}_{m},{v}_{m})& =& R\frac{\partial {u}_{m}\left(1)}{\partial t}\frac{\partial {v}_{m}\left(1)}{\partial t}.\end{array}Let us denote by Uh{U}_{h}a piecewise cubic Hermite interpolation function space. Define the approximation space Sh(m)=Uh∩H0,ω,m2(I){S}_{h}\left(m)={U}_{h}\cap {H}_{0,\omega ,m}^{2}\left(I). Then the discrete variational form associated with (3.1) is: Find (λmh,umh≠0)∈R×Sh(m)\left({\lambda }_{mh},{u}_{mh}\ne 0)\in {\mathbb{R}}\times {S}_{h}\left(m), such that (3.2)Am(umh,vmh)=λmhBm(umh,vmh),∀vmh∈Sh(m).{A}_{m}\left({u}_{mh},{v}_{mh})={\lambda }_{mh}{B}_{m}\left({u}_{mh},{v}_{mh}),\hspace{1em}\forall {v}_{mh}\in {S}_{h}\left(m).4Error estimation of approximation solutionsFor the sake of brevity, we shall use the expression a≲ba\lesssim bwhich denotes a≤cba\le cb, where ccis a positive constant.Lemma 1For any um,vm∈H0,ω,m2(I){u}_{m},{v}_{m}\in {H}_{0,\omega ,m}^{2}\left(I), the following equalities hold: (4.1)∫I(t+1)LmumLmvmdt=∫I(t+1)um″vm″dt+(2m2+1)∫I1t+1um′vm′dt+m2(m2−4)∫I1(t+1)3umvmdt+um′(1)vm′(1)\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\left(2{m}^{2}+1)\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t\\ & & +{m}^{2}\left({m}^{2}-4)\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+{u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1)\end{array}with m≠1m\ne 1, and(4.2)∫I(t+1)LmumLmvmdt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1umvm′−mt+1vmdt+(1+m)um′(1)vm′(1)\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t\\ & & +{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m){u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1)\end{array}with m=1m=1.ProofUsing integration by parts, pole conditions, and boundary conditions, we derive that (4.3)∫I(um″vm′+um′vm″)dx=um′(1)vm′(1),\mathop{\int }\limits_{I}\left({u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime} }+{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime\prime} }){\rm{d}}x={u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1),(4.4)∫I1(t+1)2(um′vm+umvm′)dt=2∫I1(t+1)3umvmdt,\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}\left({u}_{m}^{^{\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime} }){\rm{d}}t=2\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t,(4.5)∫I1t+1(um″vm+umvm″)dt=2∫I1(t+1)3umvmdt−2∫I1(t+1)um′vm′dt.\mathop{\int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime\prime} }){\rm{d}}t=2\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t-2\mathop{\int }\limits_{I}\frac{1}{\left(t+1)}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t.Then when m≠1m\ne 1, we derive from (4.3)–(4.5) that ∫I(t+1)LmumLmvmdt=∫I(t+1)um″+um′−m2t+1umvm″+vm′t+1−m2(t+1)2vmdt=∫I(t+1)um″vm″dt+∫Ium″vm′+um′vm″dt−m2∫I1(t+1)2(um′vm+umvm′)dt−m2∫I1t+1(um″vm+umvm″)dt+m4∫I1(t+1)3umvmdt+∫I1(t+1)um′vm′dt=∫I(t+1)um″vm″dt+(2m2+1)∫I1t+1um′vm′dt+m2(m2−4)∫I1(t+1)3umvmdt+um′(1)vm′(1).\hspace{-1em}\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left[\left(t+1){u}_{m}^{^{\prime\prime} }+{u}_{m}^{^{\prime} }-\frac{{m}^{2}}{t+1}{u}_{m}\right]\left[{v}_{m}^{^{\prime\prime} }+\frac{{v}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{v}_{m}\right]{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}{u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime} }+{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t-{m}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}\left({u}_{m}^{^{\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime} }){\rm{d}}t\\ & & -{m}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime\prime} }{v}_{m}+{u}_{m}{v}_{m}^{^{\prime\prime} }){\rm{d}}t+{m}^{4}\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{\left(t+1)}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){u}_{m}^{^{\prime\prime} }{v}_{m}^{^{\prime\prime} }{\rm{d}}t+\left(2{m}^{2}+1)\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{u}_{m}^{^{\prime} }{v}_{m}^{^{\prime} }{\rm{d}}t+{m}^{2}\left({m}^{2}-4)\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{u}_{m}{v}_{m}{\rm{d}}t+{u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1).\end{array}When m=1m=1, we have ∫I(t+1)LmumLmvmdt=∫I(t+1)um″+um′t+1−m2(t+1)2umvm″+vm′t+1−m2(t+1)2vmdt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1um×vm′−mt+1vmdt+(1+m)∫Ium′−mt+1umvm′−mt+1vm′dt=∫I(t+1)um′−mt+1um′vm′−mt+1vm′dt+(1+m)2∫I1t+1um′−mt+1um×vm′−mt+1vmdt+(1+m)um′(1)vm′(1).□\begin{array}{rcl}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1)\left[{u}_{m}^{^{\prime\prime} }+\frac{{u}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{u}_{m}\right]\left[{v}_{m}^{^{\prime\prime} }+\frac{{v}_{m}^{^{\prime} }}{t+1}-\frac{{m}^{2}}{{\left(t+1)}^{2}}{v}_{m}\right]{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t+{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\\ & & \times \left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m)\mathop{\displaystyle \int }\limits_{I}{\left[\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)\right]}^{^{\prime} }{\rm{d}}t\\ & =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)}^{^{\prime} }{\left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right)}^{^{\prime} }{\rm{d}}t+{\left(1+m)}^{2}\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}\left({u}_{m}^{^{\prime} }-\frac{m}{t+1}{u}_{m}\right)\\ & & \times \left({v}_{m}^{^{\prime} }-\frac{m}{t+1}{v}_{m}\right){\rm{d}}t+\left(1+m){u}_{m}^{^{\prime} }\left(1){v}_{m}^{^{\prime} }\left(1).\hspace{18em}\square \end{array}Theorem 1Am(um,vm){A}_{m}\left({u}_{m},{v}_{m})is a bounded and coercive bilinear functional on H0,ω,m2(I)×H0,ω,m2(I){H}_{0,\omega ,m}^{2}\left(I)\times {H}_{0,\omega ,m}^{2}\left(I), i.e., ∣Am(um,vm)∣≲‖um‖2,ω,m‖vm‖2,ω,m,| {A}_{m}\left({u}_{m},{v}_{m})| \lesssim \Vert {u}_{m}{\Vert }_{2,\omega ,m}\Vert {v}_{m}{\Vert }_{2,\omega ,m},Am(um,um)≳‖um‖2,ω,m2.{A}_{m}\left({u}_{m},{u}_{m})\gtrsim \Vert {u}_{m}{\Vert }_{2,\omega ,m}^{2}.ProofFrom Cauchy-Schwarz inequality, we derive that ∣Am(um,vm)∣=∫I(t+1)LmumLmvmdt≤∫I(t+1)∣Lmum∣2dt12∫I(t+1)∣Lmvm∣2dt12≲‖um‖2,ω,m‖v‖2,ω,m,Am(um,um)=∫I(t+1)∣Lmum∣2dt≳‖um‖2,ω,m2.□\hspace{7.8em}\begin{array}{rcl}| {A}_{m}\left({u}_{m},{v}_{m})| & =& \left|\hspace{-0.33em}\mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{m}{u}_{m}{L}_{m}{v}_{m}{\rm{d}}t\hspace{-0.33em}\right|\\ & \le & {\left[\mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{u}_{m}{| }^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}{\left[\mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{v}_{m}{| }^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}\\ & \lesssim & \Vert {u}_{m}{\Vert }_{2,\omega ,m}\Vert v{\Vert }_{2,\omega ,m},\\ {A}_{m}\left({u}_{m},{u}_{m})& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1)| {L}_{m}{u}_{m}{| }^{2}{\rm{d}}t\gtrsim \Vert {u}_{m}{\Vert }_{2,\omega ,m}^{2}.\hspace{18em}\square \end{array}Lemma 3.2. Let λml{\lambda }_{m}^{l}be the lth eigenvalue of the variational form (3.1). Use Vl{V}_{l}to denote any ll-dimensional subspace of H0,ω,m2(I){H}_{0,\omega ,m}^{2}\left(I). For λm1≤λm2≤⋯≤λml≤⋯{\lambda }_{m}^{1}\le {\lambda }_{m}^{2}\hspace{0.33em}\hspace{0.33em}\le \cdots \le {\lambda }_{m}^{l}\hspace{0.33em}\le \cdots , it holds(4.6)λml=minVl⊂H0,ω,m2(I)maxvm∈VlAm(vm,vm)Bm(vm,vm).{\lambda }_{m}^{l}=\mathop{\min }\limits_{{V}_{l}\subset {H}_{0,\omega ,m}^{2}\left(I)}\mathop{\max }\limits_{{v}_{m}\in {V}_{l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}.ProofSee Theorem 3.1 in [17].□Lemma 3.3. Let λmi{\lambda }_{m}^{i}be the eigenvalue of the variational form (3.1) and be arranged in the ascending order. DefineWi,j=span{umi,…,umj},{W}_{i,j}={\rm{span}}\{{u}_{m}^{i},\ldots ,{u}_{m}^{j}\},where umi{u}_{m}^{i}is the eigenfunction associated with λmi{\lambda }_{m}^{i}. Then there hold(4.7)λml=maxvm∈Wk,lAm(vm,vm)Bm(vm,vm)k≤l,{\lambda }_{m}^{l}=\mathop{\max }\limits_{{v}_{m}\in {W}_{k,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\hspace{1em}k\le l,(4.8)λml=minvm∈Wl,nAm(vm,vm)Bm(vm,vm)l≤n.{\lambda }_{m}^{l}=\mathop{\min }\limits_{{v}_{m}\in {W}_{l,n}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\hspace{1em}l\le n.ProofSee Lemma 3.2 in [17].□For the discrete form (3.2), the following minimax principle is also effective (see [17]).Lemma 3.4. Let λmhl{\lambda }_{mh}^{l}be the eigenvalue of the discrete variational form (3.2). Use Vlh{V}_{lh}to denote any ll-dimensional subspace of Sh(m){S}_{h}\left(m). For λmh1≤λmh2≤⋯≤λmhl≤⋯{\lambda }_{mh}^{1}\le {\lambda }_{mh}^{2}\hspace{0.33em}\le \cdots \le {\lambda }_{mh}^{l}\hspace{0.25em}\le \cdots , it holds(4.9)λmhl=minVhl⊂Sh(m)maxvm∈VhlAm(vm,vm)Bm(vm,vm).{\lambda }_{mh}^{l}=\mathop{\min }\limits_{{V}_{hl}\subset {S}_{h}\left(m)}\mathop{\max }\limits_{{v}_{m}\in {V}_{hl}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}.Define an orthogonal projection Qh2,m:H0,ω,m2(I)→Sh(m){Q}_{h}^{2,m}:{H}_{0,\omega ,m}^{2}\left(I)\to {S}_{h}\left(m)by Am(um−Qh2,mum,vmh)=0,∀vmh∈Sh(m).{A}_{m}\left({u}_{m}-{Q}_{h}^{2,m}{u}_{m},{v}_{mh})=0,\hspace{1em}\forall {v}_{mh}\in {S}_{h}\left(m).Theorem 2Let λmhl{\lambda }_{mh}^{l}be the approximation solution of λml{\lambda }_{m}^{l}. Then it holds(4.10)0<λml≤λmhl≤λmlmaxvm∈W1,lBm(vm,vm)Bm(Qh2,mvm,Qh2,mvm).0\lt {\lambda }_{m}^{l}\le {\lambda }_{mh}^{l}\le {\lambda }_{m}^{l}\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.ProofAccording to the positive definite property of Am(um,vm){A}_{m}\left({u}_{m},{v}_{m})and Bm(um,vm){B}_{m}\left({u}_{m},{v}_{m})we derive that λml>0{\lambda }_{m}^{l}\gt 0. Since Sh(m)⊂H0,ω,m2(I){S}_{h}\left(m)\subset {H}_{0,\omega ,m}^{2}\left(I), then from (4.6) and (4.9) we obtain λml≤λmhl{\lambda }_{m}^{l}\le {\lambda }_{mh}^{l}. Let Qh2,mW1,l{Q}_{h}^{2,m}{W}_{1,l}be the space spanned by Qh2,mum1,Qh2,mum2,…,Qh2,muml{Q}_{h}^{2,m}{u}_{m}^{1},{Q}_{h}^{2,m}{u}_{m}^{2},\ldots ,{Q}_{h}^{2,m}{u}_{m}^{l}. From the statements of Lemma 4.1 in [17], we know that Qh2,mW1,l{Q}_{h}^{2,m}{W}_{1,l}is an ll-dimensional subspace of Sh(m){S}_{h}\left(m). We derive from the minimax principle that λmhl≤maxvm∈Qh2,mW1,lAm(vm,vm)Bm(vm,vm)=maxvm∈W1,lAm(Qh2,mvm,Qh2,mvm)Bm(Qh2,mvm,Qh2,mvm).{\lambda }_{mh}^{l}\le \mathop{\max }\limits_{{v}_{m}\in {Q}_{h}^{2,m}{W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}=\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.From the bilinear property of Am(vm,vm){A}_{m}\left({v}_{m},{v}_{m}), we have Am(vm,vm)=Am(Qh2,mvm,Qh2,mvm)+2Am(vm−Qh2,mvm,Qh2,mvm)+Am(vm−Qh2,mvm,vm−Qh2,mvm){A}_{m}\left({v}_{m},{v}_{m})={A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})+2{A}_{m}({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})+{A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{v}_{m}-{Q}_{h}^{2,m}{v}_{m}). Further from Am(vm−Qh2,mvm,Qh2,mvm)=0{A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})=0and the non-negativity of Am(vm−Qh2,mvm,vm−Qh2,mvm){A}_{m}\left({v}_{m}-{Q}_{h}^{2,m}{v}_{m},{v}_{m}-{Q}_{h}^{2,m}{v}_{m}), we have Am(Qh2,mvm,Qh2,mvm)≤Am(vm,vm).{A}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})\le {A}_{m}\left({v}_{m},{v}_{m}).Thus, we obtain that λmhl≤maxvm∈W1,lAm(vm,vm)Bm(Qh2,mvm,Qh2,mvm)=maxvm∈W1,lAm(vm,vm)Bm(vm,vm)Bm(vm,vm)Bm(Qh2,mvm,Qh2,mvm)≤λmlmaxvm∈W1,lBm(vm,vm)Bm(Qh2,mvm,Qh2,mvm).\begin{array}{rcl}{\lambda }_{mh}^{l}& \le & \mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}\\ & =& \mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{A}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({v}_{m},{v}_{m})}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}\\ & \le & {\lambda }_{m}^{l}\mathop{\max }\limits_{{v}_{m}\in {W}_{1,l}}\frac{{B}_{m}\left({v}_{m},{v}_{m})}{{B}_{m}\left({Q}_{h}^{2,m}{v}_{m},{Q}_{h}^{2,m}{v}_{m})}.\end{array}The proof is complete.□Define the interpolation operator Imh:H0,ω,m2(I)→Uh{I}_{mh}:{H}_{0,\omega ,m}^{2}\left(I)\to {U}_{h}by Imhum(t)=H3,m,i(t),t∈Ii,{I}_{mh}{u}_{m}\left(t)={H}_{3,m,i}\left(t),\hspace{1em}t\in {I}_{i},where Ii=[ti−1,ti]{I}_{i}=\left[{t}_{i-1},{t}_{i}], H3,mi(t){H}_{3,mi}\left(t)is a cubic Hermite interpolation polynomial of um{u}_{m}in Ii{I}_{i}. Let umi(t)=um(t),t∈Ii.{u}_{mi}\left(t)={u}_{m}\left(t),\hspace{1em}t\in {I}_{i}.From the remainder theorem of cubic Hermite interpolation, we have umi(t)−H3,m,i(t)=(umi)(4)(ξmi(t))4!(t−ti−1)2(t−ti)2,{u}_{mi}\left(t)-{H}_{3,m,i}\left(t)=\frac{{\left({u}_{mi})}^{\left(4)}\left({\xi }_{mi}\left(t))}{4\&#x0021;}{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2},where ξmi(t)∈Ii{\xi }_{mi}\left(t)\in {I}_{i}is a function depending on tt.Theorem 3Let Emi(t)=(umi)(4)(ξmi(t))4!{E}_{mi}\left(t)=\frac{{\left({u}_{mi})}^{\left(4)}\left({\xi }_{mi}\left(t))}{4\&#x0021;}, um∈H0,ω,m2(I){u}_{m}\in {H}_{0,\omega ,m}^{2}\left(I). Assume that um{u}_{m}is sufficiently smooth such that ∣∂tkEmi(t)∣≤M(k=0,1,2)| {\partial }_{t}^{k}{E}_{mi}\left(t)| \le M\left(k=0,1,2), where M is a constant greater than zero. Then the following inequality holds: (4.11)‖∂t2(Imhum−um)‖≲h2,\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m})\Vert \lesssim {h}^{2},where h=max1≤i≤n{hi}h={\max }_{1\le i\le n}\left\{{h}_{i}\right\}, hi=ti−ti−1{h}_{i}={t}_{i}-{t}_{i-1}, ‖um‖=∫Ium2dt12\Vert {u}_{m}\Vert ={\left[{\int }_{I}{u}_{m}^{2}{\rm{d}}t\right]}^{\tfrac{1}{2}}.ProofSince umi(t)−H3,m,i(t)=Emi(t)(t−ti−1)2(t−ti)2,{u}_{mi}\left(t)-{H}_{3,m,i}\left(t)={E}_{mi}\left(t){\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2},then we have ∂t2(umi(t)−H3,m,i(t))=∂t2Emi(t)(t−ti−1)2(t−ti)2+2∂tEmi(t)∂t[(t−ti−1)2(t−ti)2]+Emi(t)∂t2[(t−ti−1)2(t−ti)2].{\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t))={\partial }_{t}^{2}{E}_{mi}\left(t){\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}+2{\partial }_{t}{E}_{mi}\left(t){\partial }_{t}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]+{E}_{mi}\left(t){\partial }_{t}^{2}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}].Thus, we obtain ∣∂t2(umi(t)−H3,m,i(t))∣2≲[(t−ti−1)2(t−ti)2]2+[∂t((t−ti−1)2(t−ti)2)]2+{∂t2[(t−ti−1)2(t−ti)2]}2=[(t−ti−1)(t−ti)]4+4[(t−ti−1)(t−ti)(2t−ti−1−ti)]2+4[4(t−ti−1)(t−ti)+(t−ti−1)2+(t−ti)2]2≤hi28+4hihi222+8hi22+hi22≲hi4.\begin{array}{rcl}| {\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t)){| }^{2}& \lesssim & {\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]}^{2}+{\left[{\partial }_{t}\left({\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2})]}^{2}+{\left\{{\partial }_{t}^{2}\left[{\left(t-{t}_{i-1})}^{2}{\left(t-{t}_{i})}^{2}]\right\}}^{2}\\ & =& {\left[\left(t-{t}_{i-1})\left(t-{t}_{i})]}^{4}+4{\left[\left(t-{t}_{i-1})\left(t-{t}_{i})\left(2t-{t}_{i-1}-{t}_{i})]}^{2}+4{\left[4\left(t-{t}_{i-1})\left(t-{t}_{i})+{\left(t-{t}_{i-1})}^{2}+{\left(t-{t}_{i})}^{2}]}^{2}\\ & \le & {\left(\frac{{h}_{i}}{2}\right)}^{8}+4{\left[{h}_{i}{\left(\frac{{h}_{i}}{2}\right)}^{2}\right]}^{2}+{\left[8{\left(\frac{{h}_{i}}{2}\right)}^{2}+{h}_{i}^{2}\right]}^{2}\lesssim {h}_{i}^{4}.\end{array}Thus, ‖∂t2(Imhum−um)‖2=∑i=1n∫Ii[∂t2(umi(t)−H3,m,i(t))]2dt≲∑i=1nhi5≲h4.\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m}){\Vert }^{2}=\mathop{\sum }\limits_{i=1}^{n}\mathop{\int }\limits_{{I}_{i}}{\left[{\partial }_{t}^{2}\left({u}_{mi}\left(t)-{H}_{3,m,i}\left(t))]}^{2}{\rm{d}}t\lesssim \mathop{\sum }\limits_{i=1}^{n}{h}_{i}^{5}\lesssim {h}^{4}.Furthermore, we have ‖∂t2(Imhum−um)‖≲h2.\Vert {\partial }_{t}^{2}\left({I}_{mh}{u}_{m}-{u}_{m})\Vert \lesssim {h}^{2}.The proof is complete.□Theorem 4Let λmhl{\lambda }_{mh}^{l}be the approximate eigenvalue of λml{\lambda }_{m}^{l}. Assume that um∈H0,ω,m2(I){u}_{m}\in {H}_{0,\omega ,m}^{2}\left(I)and satisfies the condition of Theorem 3, then the following inequality holds: ∣λmhl−λml∣≲h4,| {\lambda }_{mh}^{l}-{\lambda }_{m}^{l}| \lesssim {h}^{4},where c(l)c\left(l)is a constant independent of h.ProofFor brief, we only give the proof for the case of m≠1m\ne 1, and it can be similarly proven for the case of m=1m=1. For ∀q∈W1,l\forall \hspace{-0.25em}q\in {W}_{1,l}, we have q=∑i=1lqiumiq={\sum }_{i=1}^{l}{q}_{i}{u}_{m}^{i}. By using the orthogonality of the characteristics function umi{u}_{m}^{i}and Bm(umi,umi)=1{B}_{m}\left({u}_{m}^{i},{u}_{m}^{i})=1, we have Bm(q,q)−Bm(Qh2,mq,Qh2,mq)Bm(q,q)≤2∣Bm(q,q−Qh2,mq)∣Bm(q,q)≤2∑i,j=1l∣qiqjBm(umi−Qh2,mumi,umj)∣∑i=1l∣qi∣2≤2lmaxi,j=1,…,l∣Bm(umi−Qh2,mumi,umj)∣.\begin{array}{rcl}\frac{{B}_{m}\left(q,q)-{B}_{m}\left({Q}_{h}^{2,m}q,{Q}_{h}^{2,m}q)}{{B}_{m}\left(q,q)}& \le & \frac{2| {B}_{m}\left(q,q-{Q}_{h}^{2,m}q)| }{{B}_{m}\left(q,q)}\\ & \le & \frac{2{\displaystyle \sum }_{i,j=1}^{l}| {q}_{i}{q}_{j}{B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| }{{\displaystyle \sum }_{i=1}^{l}| {q}_{i}{| }^{2}}\\ & \le & 2l\mathop{\max }\limits_{i,j=1,\ldots ,l}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| .\end{array}From Cauchy-Schwarz inequality we have ∣Bm(umi−Qh2,mumi,umj)∣=1λmj∣λmjBm(umj,umi−Qh2,mumi)∣=1λmj∣Am(umj,umi−Qh2,mumi)∣=1λmj∣Am(umj−Qh2,mumj,umi−Qh2,mumi)∣≤1λmj[Am(umj−Qh2,mumj,umj−Qh2,mumj)]12[Am(umi−Qh2,mumi,umi−Qh2,mumi)]12.\begin{array}{rcl}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| & =& \frac{1}{{\lambda }_{m}^{j}}| {\lambda }_{m}^{j}{B}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & =& \frac{1}{{\lambda }_{m}^{j}}| {A}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & =& \frac{1}{{\lambda }_{m}^{j}}| {A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j})]}^{\tfrac{1}{2}}{\left[{A}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})]}^{\tfrac{1}{2}}.\end{array}When m≥2m\ge 2, from Hardy inequality (cf. B8.6 in [16]) we derive ∫I1(t+1)3(umi)2dt≲∫I1t+1(∂tumi)2dt,\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\left({u}_{m}^{i})}^{2}{\rm{d}}t\lesssim \mathop{\int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t,\hspace{3.2em}∫I1(t+1)2(∂tumi)2dt≲∫I(∂t2umi)2dt.\mathop{\int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\lesssim \mathop{\int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t.\hspace{4.7em}Since [∂tumi(1)]2=14∫I∂t((t+1)∂tumi)dt2=14∫I∂tumi+(t+1)∂t2umidt2≲∫I(∂tumi)2dt+∫I(t+1)(∂t2umi)2dt≲∫I1t+1(∂tumi)2dt+∫I(t+1)(∂t2umi)2dt,\begin{array}{rcl}{\left[{\partial }_{t}{u}_{m}^{i}\left(1)]}^{2}& =& \frac{1}{4}{\left[\mathop{\displaystyle \int }\limits_{I}{\partial }_{t}\left(\left(t+1){\partial }_{t}{u}_{m}^{i}){\rm{d}}t\right]}^{2}\\ & =& \frac{1}{4}{\left[\mathop{\displaystyle \int }\limits_{I}{\partial }_{t}{u}_{m}^{i}+\left(t+1){\partial }_{t}^{2}{u}_{m}^{i}{\rm{d}}t\right]}^{2}\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t,\end{array}\hspace{8.15em}from Lemma 1 we have ‖umi‖2,ω,m2≲∫I(t+1)(∂t2umi)2dt+∫I1t+1(∂tumi)2dt+∫I1(t+1)3(umi)2dt≲∫I(∂t2umi)2dt+∫I1t+1(∂tumi)2dt≲∫I(∂t2umi)2dt+∫I1(t+1)2(∂tumi)2dt≲∫I(∂t2umi)2dt.\begin{array}{rcl}\Vert {u}_{m}^{i}{\Vert }_{2,\omega ,m}^{2}& \lesssim & \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\left({u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t+\mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{2}}{\left({\partial }_{t}{u}_{m}^{i})}^{2}{\rm{d}}t\\ & \lesssim & \mathop{\displaystyle \int }\limits_{I}{\left({\partial }_{t}^{2}{u}_{m}^{i})}^{2}{\rm{d}}t.\end{array}Then we derive that ∣Bm(umi−Qh2,mumi,umj)∣=1λmj∣λmjBm(umj,umi−Qh2,mumi)∣≤1λmj[Am(umj−Qh2,mumj,umj−Qh2,mumj)]12[Am(umi−Qh2,mumi,umi−Qh2,mumi)]12≤1λmj[Am(umj−Imhumj,umj−Imhumj)]12⋅[Am(umi−Imhumi,umi−Imhumi)]12≤Mλmj‖umj−Ihumj‖2,ω,m‖umi−Ihumi‖2,ω,m≲∫I[∂t2(umj−Imhumj)]2dt12∫I[∂t2(umi−Imhumi)]2dt12≲‖∂t2(umj−Imhumj)‖⋅‖∂t2(umi−Imhumi)‖.\begin{array}{rcl}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| & =& \frac{1}{{\lambda }_{m}^{j}}| {\lambda }_{m}^{j}{B}_{m}\left({u}_{m}^{j},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})| \\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j},{u}_{m}^{j}-{Q}_{h}^{2,m}{u}_{m}^{j})]}^{\tfrac{1}{2}}{\left[{A}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i})]}^{\tfrac{1}{2}}\\ & \le & \frac{1}{{\lambda }_{m}^{j}}{\left[{A}_{m}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j},{u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})]}^{\tfrac{1}{2}}\cdot {\left[{A}_{m}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i},{u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})]}^{\tfrac{1}{2}}\\ & \le & \frac{M}{{\lambda }_{m}^{j}}\Vert {u}_{m}^{j}-{I}_{h}{u}_{m}^{j}{\Vert }_{2,\omega ,m}\Vert {u}_{m}^{i}-{I}_{h}{u}_{m}^{i}{\Vert }_{2,\omega ,m}\\ & \lesssim & {\left(\mathop{\displaystyle \int }\limits_{I}{\left[{\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})]}^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}{\left(\mathop{\displaystyle \int }\limits_{I}{\left[{\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})]}^{2}{\rm{d}}t\right)}^{\tfrac{1}{2}}\\ & \lesssim & \Vert {\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})\Vert \cdot \Vert {\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})\Vert .\end{array}Similarly, when m=0m=0, we derive that ∣Bm(umi−Qh2,mumi,umj)∣≲‖∂t2(umj−Imhumj)‖⋅‖∂t2(umi−Imhumi)‖.| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| \lesssim \Vert {\partial }_{t}^{2}\left({u}_{m}^{j}-{I}_{mh}{u}_{m}^{j})\Vert \cdot \Vert {\partial }_{t}^{2}\left({u}_{m}^{i}-{I}_{mh}{u}_{m}^{i})\Vert .Since Bm(q,q)Bm(Qh2,mq,Qh2,mq)≤11−2lmaxi,j=1,…,l∣Bm(umi−Qh2,mumi,umj)∣,\frac{{B}_{m}\left(q,q)}{{B}_{m}\left({Q}_{h}^{2,m}q,{Q}_{h}^{2,m}q)}\le \frac{1}{1-2l{\max }_{i,j=1,\ldots ,l}| {B}_{m}\left({u}_{m}^{i}-{Q}_{h}^{2,m}{u}_{m}^{i},{u}_{m}^{j})| },we obtain from Theorems 2 and 3 the desired results.□5Efficient implementation of the algorithmIn order to efficiently solve the problems (3.2), we start by constructing a set of basis functions which satisfy boundary conditions. Let φ00(t)=2t−t0h1+1t−t0h1−12,ti≤t≤ti+1,0,others,φ01(t)=h1−2(t−t0)(t−t1)2,t0≤t≤t10,others,φi0(t)=1+t−tihi21−2t−tihi,ti−1≤t≤ti,2t−tihi+1+1t−tihi+1−12,ti≤t≤ti+1,0,others,φi1(t)=hi−2(t−ti)(t−ti−1)2,ti−1≤t≤ti,hi+1−2(t−ti)(t−ti+1)2,ti≤t≤ti+1,0,others,φn1(t)=hn−2(t−tn)(t−tn−1)2,tn−1≤t≤tn,0,others,\begin{array}{rcl}{\varphi }_{0}^{0}\left(t)& =& \left\{\begin{array}{ll}\left(2\frac{t-{t}_{0}}{{h}_{1}}+1\right){\left(\frac{t-{t}_{0}}{{h}_{1}}-1\right)}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{0}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{1}^{-2}\left(t-{t}_{0}){\left(t-{t}_{1})}^{2},& {t}_{0}\le t\le {t}_{1}\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{i}^{0}\left(t)& =& \left\{\begin{array}{ll}{\left(1+\frac{t-{t}_{i}}{{h}_{i}}\right)}^{2}\left(1-2\frac{t-{t}_{i}}{{h}_{i}}\right),& {t}_{i-1}\le t\le {t}_{i},\\ \left(2\frac{t-{t}_{i}}{{h}_{i+1}}+1\right){\left(\frac{t-{t}_{i}}{{h}_{i+1}}-1\right)}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{i}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{i}^{-2}\left(t-{t}_{i}){\left(t-{t}_{i-1})}^{2},& {t}_{i-1}\le t\le {t}_{i},\\ {h}_{i+1}^{-2}\left(t-{t}_{i}){\left(t-{t}_{i+1})}^{2},& {t}_{i}\le t\le {t}_{i+1},\\ 0,& {\rm{others}},\end{array}\right.\\ {\varphi }_{n}^{1}\left(t)& =& \left\{\begin{array}{ll}{h}_{n}^{-2}\left(t-{t}_{n}){\left(t-{t}_{n-1})}^{2},& {t}_{n-1}\le t\le {t}_{n},\\ 0,& {\rm{others}},\end{array}\right.\end{array}where i=1,…,n−1i=1,\ldots ,n-1. It is clear that Sh(0)=span{φ00(t),…,φn−10(t),φ11(t),…,φn1(t)};Sh(1)=span{φ10(t),…,φn−10(t),φ01(x),…,φn1(t)};Sh(m)=span{φ10(t),…,φn−10(t),φ11(x),…,φn1(t)},(m≥2).\begin{array}{l}{S}_{h}\left(0)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{0}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{1}^{1}\left(t),\ldots ,{\varphi }_{n}^{1}\left(t)\right\};\\ {S}_{h}\left(1)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{1}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{0}^{1}\left(x),\ldots ,{\varphi }_{n}^{1}\left(t)\right\};\\ {S}_{h}\left(m)=\hspace{0.1em}\text{span}\hspace{0.1em}\left\{{\varphi }_{1}^{0}\left(t),\ldots ,{\varphi }_{n-1}^{0}\left(t),{\varphi }_{1}^{1}\left(x),\ldots ,{\varphi }_{n}^{1}\left(t)\right\},\left(m\ge 2).\end{array}Denote aijpq=∫I(t+1)(φjp)″(φiq)″dt,bijpq=∫I1t+1(φjp)′(φiq)′dt,cijpq=∫I1(t+1)3φjpφiqdt,dijpq=φjp(1)φiq(1),\begin{array}{rcl}{a}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){\left({\varphi }_{j}^{p})}^{^{\prime\prime} }{\left({\varphi }_{i}^{q})}^{^{\prime\prime} }{\rm{d}}t,\hspace{1em}{b}_{ij}^{pq}=\mathop{\displaystyle \int }\limits_{I}\frac{1}{t+1}{\left({\varphi }_{j}^{p})}^{^{\prime} }{\left({\varphi }_{i}^{q})}^{^{\prime} }{\rm{d}}t,\\ {c}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\frac{1}{{\left(t+1)}^{3}}{\varphi }_{j}^{p}{\varphi }_{i}^{q}{\rm{d}}t,\hspace{1em}{d}_{ij}^{pq}={\varphi }_{j}^{p}\left(1){\varphi }_{i}^{q}\left(1),\end{array}where p,q=0,1p,q=0,1.Next, we will derive the matrix form of the discrete variational scheme (3.2).Case 1. When m=0m=0, let (5.1)u0h=u00φ00+un1φn1+∑i=1n−1(ui0φi0+ui1φi1).{u}_{0h}={u}_{0}^{0}{\varphi }_{0}^{0}+{u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).Plugging the expression (5.1) in (3.2) and taking v0h{v}_{0h}through all the basis functions in Sh(0){S}_{h}\left(0), we derive that (5.2)(A0+B0+D0)U0=λ0hRD0U0,\left({A}_{0}+{B}_{0}+{D}_{0}){U}^{0}={\lambda }_{0h}R{D}_{0}{U}^{0},where A0=(aij00)(aij10)(aij01)(aij11),B0=(bij00)(bij10)(bij01)(bij11),D0=(dij00)(dij10)(dij01)(dij11),{A}_{0}=\left(\begin{array}{cc}\left({a}_{ij}^{00})& \left({a}_{ij}^{10})\\ \left({a}_{ij}^{01})& \left({a}_{ij}^{11})\end{array}\right),\hspace{1.0em}{B}_{0}=\left(\begin{array}{cc}\left({b}_{ij}^{00})& \left({b}_{ij}^{10})\\ \left({b}_{ij}^{01})& \left({b}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{0}=\left(\begin{array}{cc}\left({d}_{ij}^{00})& \left({d}_{ij}^{10})\\ \left({d}_{ij}^{01})& \left({d}_{ij}^{11})\end{array}\right),\hspace{1.0em}U0=(u00,…,uN−10,u11,…,un1)T.{U}^{0}={\left({u}_{0}^{0},\ldots ,{u}_{N-1}^{0},{u}_{1}^{1},\ldots ,{u}_{n}^{1})}^{T}.\hspace{0.4em}Similarly, when m=1m=1, let (5.3)u1h=u01φ01+un1φn1+∑i=1n−1(ui0φi0+ui1φi1).{u}_{1h}={u}_{0}^{1}{\varphi }_{0}^{1}+{u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).\hspace{0.5em}Plugging the expression (5.3) in (3.2) and taking v1h{v}_{1h}through all the basis functions in Sh(1){S}_{h}\left(1), we obtain (5.4)[A1+(1+m)D1]U1=λ1hRD1U1,\left[{A}_{1}+\left(1+m){D}_{1}]{U}^{1}={\lambda }_{1h}R{D}_{1}{U}^{1},where A1=(a˜ij00)(a˜ij10)(a˜ij01)(a˜ij11),D1=(d˜ij00)(d˜ij10)(d˜ij01)(d˜ij11),\hspace{4em}{A}_{1}=\left(\begin{array}{cc}\left({\tilde{a}}_{ij}^{00})& \left({\tilde{a}}_{ij}^{10})\\ \left({\tilde{a}}_{ij}^{01})& \left({\tilde{a}}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{1}=\left(\begin{array}{cc}\left({\tilde{d}}_{ij}^{00})& \left({\tilde{d}}_{ij}^{10})\\ \left({\tilde{d}}_{ij}^{01})& \left({\tilde{d}}_{ij}^{11})\end{array}\right),\hspace{2.55em}a˜ijpq=∫I(t+1)L1φjpL1φiqdt,d˜ijpq=φjp(1)φiq(1),U1=(u10…,un−10,u01,…,un1)T.\begin{array}{rcl}{\tilde{a}}_{ij}^{pq}& =& \mathop{\displaystyle \int }\limits_{I}\left(t+1){L}_{1}{\varphi }_{j}^{p}{L}_{1}{\varphi }_{i}^{q}{\rm{d}}t,\hspace{1em}{\tilde{d}}_{ij}^{pq}={\varphi }_{j}^{p}\left(1){\varphi }_{i}^{q}\left(1),\\ {U}^{1}& =& {\left({u}_{1}^{0}\ldots ,{u}_{n-1}^{0},{u}_{0}^{1},\ldots ,{u}_{n}^{1})}^{T}.\end{array}When m≥2m\ge 2, let (5.5)u2h=un1φn1+∑i=1n−1(ui0φi0+ui1φi1).\hspace{3.1em}{u}_{2h}={u}_{n}^{1}{\varphi }_{n}^{1}+\mathop{\sum }\limits_{i=1}^{n-1}\left({u}_{i}^{0}{\varphi }_{i}^{0}+{u}_{i}^{1}{\varphi }_{i}^{1}).\hspace{2.95em}Plugging the expression (5.5) in (3.2) and taking vmh{v}_{mh}through all the basis functions in Sh(m),{S}_{h}\left(m),we gain (5.6)[Am+(2m2+1)Bm+m2(m2−4)Cm+Dm]Um=λmhRDmUm,\left[{A}_{m}+\left(2{m}^{2}+1){B}_{m}+{m}^{2}\left({m}^{2}-4){C}_{m}+{D}_{m}]{U}^{m}={\lambda }_{mh}R{D}_{m}{U}^{m},where (5.7)Am=(aij00)(aij10)(aij01)(aij11),Bm=(bij00)(bij10)(bij01)(bij11),Cm=(cij00)(cij10)(cij01)(cij11),Dm=(dij00)(dij10)(dij01)(dij11){A}_{m}=\left(\begin{array}{cc}\left({a}_{ij}^{00})& \left({a}_{ij}^{10})\\ \left({a}_{ij}^{01})& \left({a}_{ij}^{11})\end{array}\right),\hspace{1.0em}{B}_{m}=\left(\begin{array}{cc}\left({b}_{ij}^{00})& \left({b}_{ij}^{10})\\ \left({b}_{ij}^{01})& \left({b}_{ij}^{11})\end{array}\right),\hspace{1.0em}\hspace{1em}{C}_{m}=\left(\begin{array}{cc}\left({c}_{ij}^{00})& \left({c}_{ij}^{10})\\ \left({c}_{ij}^{01})& \left({c}_{ij}^{11})\end{array}\right),\hspace{1.0em}{D}_{m}=\left(\begin{array}{cc}\left({d}_{ij}^{00})& \left({d}_{ij}^{10})\\ \left({d}_{ij}^{01})& \left({d}_{ij}^{11})\end{array}\right)Um=(u10,…,un−10,u11,…,un1)T.{U}^{m}={\left({u}_{1}^{0},\ldots ,{u}_{n-1}^{0},{u}_{1}^{1},\ldots ,{u}_{n}^{1})}^{T}.Note that we know from the properties of cubic hermit interpolation basis function that the stiff matrices and mass matrices in (5.4)–(5.6) are all sparse. Thus, they can be efficiently solved.6Numerical experimentsIn order to show the accuracy and convergence of the proposed algorithm, we will carry out a series of numerical tests. We operate our programs in MATLAB 2016b.Example 1We take R=1R=1and m=0,1,2,3m=0,1,2,3. The eigenvalues for different mmand hhare listed in Table 1.Table 1Eigenvalues for m=0,1,2,3m=0,1,2,3and different hhhλ0h{\lambda }_{0h}λ1h{\lambda }_{1h}λ2h{\lambda }_{2h}λ3h{\lambda }_{3h}1/82.0000000244610734.0000000730055186.0000252085831298.0003070914540561/162.0000000158994934.0000000724494046.0000015395142628.0000191017624241/322.0000000064929494.0000000849451286.0000000956013148.0000011925251611/642.0000000026608134.0000000910470356.0000000059450748.000000074519297We know from Table 1 that the eigenvalues achieve at least six-digit accuracy with h≤132h\le \frac{1}{32}for m=0,1,2,3m=0,1,2,3. In order to further show the convergence of the algorithm, we choose the numerical solutions of h=164h=\frac{1}{64}as reference solutions, and the error figures of the approximate eigenvalues λmh(m=0,1,2,3){\lambda }_{mh}\left(m=0,1,2,3)with different hhare presented in Figure 1. We observe from Figure 1 that the numerical eigenvalues are also convergent.Figure 1Errors between numerical solutions and the reference solution for R=1R=1.Example 2We take R=2R=2and m=0,1,2,3m=0,1,2,3. The eigenvalues for different mmand hhare listed in Table 2.Table 2Eigenvalues for m=0,1,2,3m=0,1,2,3and different hhhλ0h{\lambda }_{0h}λ1h{\lambda }_{1h}λ2h{\lambda }_{2h}λ3h{\lambda }_{3h}1/81.0000000122305372.0000000365027593.0000126042915654.0001535457270281/161.0000000079497472.0000000362247023.0000007697571314.0000095508812121/321.0000000032464742.0000000424725643.0000000478006574.0000005962625811/641.0000000013304062.0000000455235173.0000000029725374.000000037259649Similarly, we observe from Table 2 that the eigenvalues have at least six-digit accuracy with h≤132h\le \frac{1}{32}for m=0,1,2,3m=0,1,2,3. We still choose the numerical solutions of h=164h=\frac{1}{64}as reference solutions, the error figures of the approximate eigenvalues λmh(m=0,1,2,3){\lambda }_{mh}\left(m=0,1,2,3)with different hhare listed in Figure 2. We see from Figure 2 that the approximation eigenvalues are also convergent. Besides, in order to show the convergence rate of our algorithm more intuitively, we also plot the error figures in semilog scale in Figures 3 and 4.Figure 2Errors between numerical solutions and the reference solution for R=2R=2.Figure 3Error curves in semilog scale between the numerical solution and the reference solution for R=1R=1.Figure 4Error curves in semilog scale between the numerical solution and the reference solution for R=2R=2.Next, we shall provide a numerical example for some larger Fourier norm mm.Example 3We take R=1R=1and m=4,5,6,7m=4,5,6,7. The eigenvalues for different mmand hhare listed in Table 3.Table 3Eigenvalues for m=4,5,6,7m=4,5,6,7and different hhhhλ4h{\lambda }_{4h}λ5h{\lambda }_{5h}λ6h{\lambda }_{6h}λ7h{\lambda }_{7h}1/410.023827239467012.075633841422614.187693723539216.39313166639011/810.001490600395312.004878672365214.012611174554816.02775272290221/1610.000093110825312.000307476583914.000803468087416.00179279509841/3210.000005818033112.000019258364014.000050462262416.00011299986481/6410.000000363585112.000001204285914.000003157754116.00000707750631/12810.000000022918412.000000075660014.000000197488516.0000004427145Likewise, we observe from Table 3 that the eigenvalues have at least five-digit accuracy with h≤164h\le \frac{1}{64}for m=4,5,6,7m=4,5,6,7. We choose the numerical solutions of h=1128h=\frac{1}{128}as reference solutions, and the error figures of the approximate eigenvalues λmh(m=4,5,6,7){\lambda }_{mh}\left(m=4,5,6,7)with different hhare listed in Figure 5. We see from Figure 5 that the approximation eigenvalues are also convergent.Figure 5Errors between numerical solutions and the reference solution for m=4,5,6,7m=4,5,6,7and R=1R=1.Introducing the usual definition of convergence order: (6.1)c(h)=log2∣λh−λh2∣∣λh2−λh4∣.c\left(h)={\log }_{2}\left(\frac{| {\lambda }_{h}-{\lambda }_{\tfrac{h}{2}}| }{| {\lambda }_{\tfrac{h}{2}}-{\lambda }_{\tfrac{h}{4}}| }\right).For brevity, we shall use formula (6.1) to calculate the convergence order of the approximation eigenvalues of m=4,5,6,7m=4,5,6,7in the unit disk and list them in Table 4. We observe from Table 4 that the convergence order is about 4.Table 4Convergence order c(h)c\left(h)for the eigenvalues with m=4,5,6,7m=4,5,6,7hhλ4h{\lambda }_{4h}λ5h{\lambda }_{5h}λ6h{\lambda }_{6h}λ7h{\lambda }_{7h}1/23.99243.84593.69603.57701/43.99853.95223.89023.81501/84.00083.98733.97093.94991/164.00033.99683.99263.98721/324.00103.99973.99823.9968Finally, we plot the eigenvalue error in log-log scale to describe the algebra convergence rate in Figure 6.Figure 6The eigenvalue errors in log-log scale between numerical solutions and the reference solution for m=4,5,6,7m=4,5,6,7and R=1R=1.7ConclusionWe present in this article a novel finite element method based on a dimension reduction scheme for a fourth-order Steklov eigenvalue problem in a circular domain. The main advantage of this method is that the original problem is transformed into a series of one-dimensional problems which can be solved in parallel. Then, by introducing the polar conditions and the weighted Sobolev space, we prove the error estimates of approximation eigenvalues by using the minimax principle. The method developed in this article can be applied to more complex problems or more general polar geometric domains which will be the subject of our future endeavors.

Journal

Open Mathematics – de Gruyter

Published: Jan 1, 2022

Keywords: fourth-order Steklov eigenvalue problem; weighted Sobolev space; dimension reduction scheme; finite element approximation; error estimation; 34L15; 65M60; 97N20

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