Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion

Finite element approximations for second-order stochastic differential equation driven by... Abstract We consider finite element approximations for a one-dimensional second-order stochastic differential equation of boundary value type driven by a fractional Brownian motion with Hurst index $$H\,{\le}\, 1/2$$. We make use of a sequence of approximate solutions with the fractional noise replaced by its piecewise constant approximations to construct the finite element approximations for the equation. The error estimate of the approximations is derived through rigorous convergence analysis. 1 Introduction Many physical and engineering phenomena can be modeled by stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) when including some levels of uncertainties. The advantage of modeling with stochastic equations is that they are able to more fully capture the practical behavior of underlying models; it also means that the corresponding numerical analysis will require new tools to simulate the systems, produce the solutions and analyse the information stored within the solutions. Stochastic equations derived from fluid flows and other engineering fields are often driven by white noise. The white noise is an uncorrelated noise with delta function as its covariance. However, random fluctuations in complex systems may not be uncorrelated, i.e., they may not be white noise. Such noises are named as colored noises. As an important class of colored noises, the fractional Brownian motion (fBm)-type noise appears naturally in the modeling of many physical and social phenomena (see, e.g., Mandelbrot & Van Ness, 1968; van Wyk et al., 2015). For example, fBm is suitable in describing the widths of consecutive annual rings of a tree and the temperature at a specific place (see, e.g., Shiryaev, 1999); it can also be applied to simulate the turbulence in an incompressible fluid flow and the prices of electricity in a liberated electricity market (see, eg., Simonsen, 2003). As a centered Gaussian process, the fBm can be defined as follows. Let $$\mathscr D\,{=}\,(0,1)$$ and denote $${\overline {\mathscr D}}$$, $$\partial \mathscr D$$ the closure and the boundary of $$\mathscr D$$, respectively. The fBm $$W\,{=}\,\{W(x),\ x\in {\overline {\mathscr D}}\}$$ on $${\overline {\mathscr D}}$$ is determined by its covariance function   Cov(x,y):=E[W(x)W(y)]=x2H+y2H−|x−y|2H2x,y∈D¯. (1.1) Here, $$H\in (0,1)$$ is the so-called Hurst index. The fBms with $$H<1/2$$ and $$H>1/2$$ are significantly different both physically and mathematically. In the first two aforementioned applications, the corresponding fBms have Hurst index $$H>1/2$$. In such cases, the physical process presents an aggregation and persistent behavior. On the other hand, the Hurst index $$H$$ is less than $$1/2$$ in the last two cases, where the process is anti-persistent and may have long-range negative interactions. These two classes of fBms are separated by the standard Brownian motion whose Hurst index is $$H\,{=}\,1/2$$. Mathematically, the fBm with $$H>1/2$$ is a Gaussian process whose covariance function has a bounded variation on $${\overline {\mathscr D}}\times {\overline {\mathscr D}}$$. The stochastic integral against the fBm with $$H>1/2$$ can be viewed as a pathwise Riemann–Stieltjes integral (or Young integral) and classical methods are applicative. On the contrary, the covariance function of the fBm with $$H\,{\le}\, 1/2$$ does not have bounded variation. This posts a particular difficulty when studying SDEs or SPDEs driven by such noises. The main objective of this article is to investigate the well posedness and finite element approximations for the following second-order SDE of boundary type driven by an fBm with $$H\,{\le}\, 1/2$$:   −d2dx2u(x)+f(x,u(x)) =g(x)+W˙(x)x∈D,u(x) =0,x∈∂D. (1.2) Here, $$g: \mathscr D\rightarrow \mathbb R$$ is square integrable, $$f: \mathscr D\times\mathbb R \rightarrow \mathbb R$$ satisfies certain conditions given in Section 2 and $$W\,{=}\,\{W(x):\ x\in {\overline {\mathscr D}}\}$$ is an fBm, determined by (1.1) with $$H\,{\le}\, 1/2$$, on a filtered probability space $$(\it{\Omega}, \mathscr F, (\mathscr F_x)_{x\in {\overline {\mathscr D}}}, \mathbb P)$$. The homogenous Dirichlet boundary condition in equation (1.2) corresponds to a second-order SDE conditioned to hit a particular point at ‘time’ $$x\,{=}\,1$$. As such it is a generalization to general fractional noise of the conditioned diffusions studied in Hairer et al. (2011). On the other hand, this equation can be considered as an elliptic SPDE in one dimension. Note that one can also study homogenous Neumann boundary condition and the main results of this article are also valid. Equation (1.2) driven by the white noise, i.e., $$H\,{=}\,1/2$$, has been considered by several authors (see, e.g., Allen et al., 1998; Du & Zhang, 2002; Gyöngy & Martínez, 2006; Martínez & Sanz-Solé, 2006; Cao et al., 2007, 2015; Zhang et al., 2015). Allen et al. (1998) investigated the finite difference and finite element approximations of the linear case of equation (1.2). They proved the first-order convergence for both the finite difference and finite element approximations. The three authors in Cao et al. (2015) investigated the finite element approximations of equation (1.2) in possibly any dimensions formulated in the form of Karhunen–Loève expansions for certain Gaussian noises. For the case where $$H>1/2$$, the well posedness and finite difference approximations can be studied using the methodology of Martínez & Sanz-Solé (2006); by treating the fBm as a colored noise with a special Riesz kernel. However, the method of treating the white noise and more regular noises does not apply to fractional noise with $$H<1/2$$, since the exact solution is less regular. To the best of our knowledge, there have not been literatures studying numerical approximations for SDEs or SPDEs driven by fractional noises with $$H<1/2$$. The primary challenge in studying the finite element approximations of equation (1.2) driven by fBm with $$H<1/2$$ is threefold: (i) as a colored noise, the increments of the fBm in two disjoint intervals are not independent; (ii) the regularity of $$\dot{W}$$ with $$H<1/2$$ is very low, and (iii) the approach of Karhunen–Loève expansions used in Cao et al. (2015) fails. In this article, we study the well posedness and the finite element approximations of equation (1.2) through a special Itô isometry which is only valid for $$H\,{\le}\, 1/2$$ (see, (2.6)). Using this isometry, we obtain the existence of a unique solution for equation (1.2) by analysing the convergence of a sequence of approximate solutions of SPDEs with the fractional noise replaced by a sum of tensor products between correlated Gaussian random variables and piecewise constant functions in the physical domain. Following the well-posedness analysis, we construct the finite element approximations of equation (1.2) through two steps. In the first step, we derive an error estimate between the exact solution and its approximations that are used in the well-posedness analysis. This error estimate also heavily depends on the aforementioned Itô isometry. In the second step, we apply the Galerkin finite element method to the approximate noise-driven SPDE, and obtain the overall error estimate of the finite element solution through an finite element error estimate for the approximate SPDE. The article is organized as follows. First, we define the weak solution and mild solution of equation (1.2) and establish their existence and uniqueness in Section 2. Next in Section 3, we derive the error estimate between the exact solution of equation (1.2) and the solution of the approximate SPDE. In Section 4, we apply a finite element method to this approximate SPDE and derive the overall error estimate of the finite element solution. Finally, a few concluding remarks are given in Section 5. We conclude this section by introducing some notations that will be used throughout the article. Denote by $$L^2(\mathscr D)$$ the space of square integrable functions in $$\mathscr D$$ with its inner product and norm denoted by $$(\cdot,\cdot)$$ and $$\|\cdot\|$$, respectively. For $$r>0$$, we use $$H^r(\mathscr D)$$ to denote the usual Sobolev space whose norm is denoted by $$\|\cdot\|_r$$. We also use $$H^1_0(\mathscr D)$$ to denote the subspace of $$H^1(\mathscr D)$$ whose elements vanish on $$\partial \mathscr D$$. We denote by $$C$$ a generic positive constant independent of either the truncation number $$n$$ or the grid size $$h$$, which will change from one line to another. 2. Well posedness of the problem In this section, we define the weak solution and mild solution of equation (1.2), and then establish their equivalence, existence and uniqueness. Definition 2.1 An $$\mathscr F_x$$-adapted stochastic process $$u\,{=}\,\{u(x):x\in \mathscr D\}$$ is called a weak solution of equation (1.2), if for every $$\phi\in C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$ vanishing on $$\partial \mathscr D$$ it holds a.s. that   −∫Du(x)ϕ″(x)dx+∫Df(x,u(x))ϕ(x)dx=∫Dg(x)ϕ(x)dx+∫Dϕ(x)dW(x). (2.1) Definition 2.2 An $$\mathscr F_x$$-adapted stochastic process $$u\,{=}\,\{u(x):x\in \mathscr D\}$$ is called a mild solution of equation (1.2), if for all $$x\in \mathscr D$$ it holds a.s. that   u(x)+∫DG(x,y)f(y,u(y))dy=∫DG(x,y)g(y)dy+∫DG(x,y)dW(y), (2.2) where $$G$$ is the Green’s function associated with the Poisson equation with Dirichlet boundary. It is well known that the related Green’s function $$G$$ is given by $$G(x,y)\,{=}\,x\wedge y-xy$$$$x,y\in {\overline {\mathscr D}}$$. Obviously, $$G$$ is Lipschitz continuous over $${\overline {\mathscr D}}\times {\overline {\mathscr D}}$$. Without loss of generality, we assume that $$f(x,0)\,{=}\,0$$ for any $$x\in \mathscr D$$. Otherwise, we simply replace $$f(x,r)$$ by $$f(x,r)-f(x,0)$$ and $$g(x)$$ by $$g(x)-f(x,0)$$. Assume furthermore that $$f$$ satisfies the following assumptions. Assumption 2.3 (1) (Monotone-type condition) There exists a positive constant $$L<\gamma$$ such that   (f(x,r)−f(x,s),r−s)≥−L|r−s|2∀ x∈D, r,s∈R, (2.3) where $$\gamma$$ is the positive constant in the Poincaré inequality (see, e.g., Adams & Fournier, 2003, Theorem 6.30):   ‖ddxv‖2≥γ‖v‖2∀ v∈H01(D). (2.4) (2) (Linear growth condition) There exists a positive constant $$\beta$$ such that   |f(x,r)−f(x,s)|≤β(1+|r−s|)∀ x∈D, r,s∈R. (2.5) We remark that these two conditions can be satisfied when $$f$$ is a sum of a nondecreasing bounded function and a Lipschitz continuous function with the Lipschitz constant less than $$\gamma$$ (see, e.g., Buckdahn & Pardoux, 1990; Gyöngy & Martínez, 2006). In the case, $$\mathscr D\,{=}\,(0,1),$$ it can be easily shown that $$\gamma \,{=}\,2$$. Therefore, we assume that $$L<2$$ throughout the rest of this article. Before establishing the well posedness of equation (1.2), we follow the approach of Bardina & Jolis (2006) to define stochastic integral with respect to the fBm $$W$$ with $$H<1/2$$. To this end, we introduce the set $$\Phi$$ of all step functions on $$\mathscr D$$ of the form   f=∑j=0N−1fjχ(aj,aj+1], where $$0\,{=}\,a_0<a_1<\cdots<a_N\,{=}\,1$$ is a partition of $$\mathscr D$$ and $$f_j\in\mathbb R$$, $$j\,{=}\,0,1,\cdots,N-1$$, $$N\in \mathbb N_+$$. For $$f\in \it{\Phi}$$, we define its integral with respect to $$W$$ by Riemann sum as   I(f)=∑j=0N−1fj(W(aj+1)−W(aj)), and for $$f,g\in \it{\Phi}$$, we define their scalar product as   Ψ(f,g):=E[I(f)I(g)]. Next we extend $$\it{\Phi}$$ through completion to a Hilbert space, denoted by $$\it{\Phi}^H$$. By Bardina & Jolis (2006, Lemma 2.1), we have a characterization of $$\it{\Phi}^H$$ through Itô isometry for simple functions:   Ψ(f,g) =H(1−2H)2∫D∫D(f(x)−f(y))(g(x)−g(y))|x−y|2−2Hdxdy +H∫Df(x)g(x)(x2H−1+(1−x)2H−1)dx∀ f,g∈Φ. (2.6) This shows that   ΦH={f∈L2(D¯):∫R∫R|f¯(x)−f¯(y)|2|x−y|2−2Hdxdy<∞}, where $$\overline{f}(x)\,{=}\,f(x)$$ when $$x\in{\overline {\mathscr D}}$$ and $$\overline{f}(x)\,{=}\,0$$ otherwise. As a consequence, the integral $$I$$ for a measurable deterministic function $$f:{\overline {\mathscr D}}\rightarrow \mathbb R$$ with respect to the fBm $$W$$ is an isometry between $$\it{\Phi}^H$$ and a subspace of $$L^2(\mathbb P)$$. Lemma 2.4 (1) The stochastic process $$\{v(x)\,{:=}\,\int_{\mathscr D} G(x,y)dW(y),\ x\in{\overline {\mathscr D}}\}$$ possesses an a.s. continuous modification. (2) Definitions 2.1 and 2.2 are equivalent to each other. Proof. Let $$x_1,x_2\in \mathscr D$$. The Ito’s isometry (2.6) yields   E[|v(x1)−v(x2)|2] =H(1−2H)2∫D∫D|[G(x1,y)−G(x2,y)]−[G(x1,z)−G(x2,z)]|2|y−z|2−2Hdydz +H∫D|G(x1,y)−G(x2,y)|2(y2H−1+(1−y)2H−1)dx. Since $$G\,{=}\,\{G(x,y):\ x,y\in {\overline {\mathscr D}}\}$$ is Lipschitz continuous with respect to both $$x$$ and $$y$$, we have   |[G(x1,y)−G(x2,y)]−[G(x1,z)−G(x2,z)]|2≤2|x1−x2|×2|y−z|. Direct calculations yield that   ∫D∫D|y−z|2H−1dydz=H(1+2H). Therefore, there exists $$C\,{=}\,C(H)$$ such that   E[|v(x1)−v(x2)|2]≤C|x1−x2|x1,x2∈D¯, from which, and the fact that $$v$$ is Gaussian, we conclude that $$v$$ has an a.s. continuous modification (see, e.g., Khoshnevisan, 2009, Exercise 4.9). Assume that $$u$$ satisfies (2.2) and let $$\phi\in C_0^\infty(\mathscr D)$$. Multiplying (2.2) by $$\phi''(x)$$, integrating over $$\mathscr D$$ and using the identity $$-\int_{\mathscr D} G(x,y) \phi''(y)\,{\rm d}y\,{=}\,\phi(x)$$, we obtain (2.1) for smooth $$\phi$$. The general case follows from the fact that $$C_0^\infty(\mathscr D)$$ is dense in $$C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$. Suppose now that $$u$$ satisfies (2.1). Choose $$\phi(x)\,{=}\,-\int_{\mathscr D} G(x,y)\psi(y)\,{\rm d}y$$ with $$\psi\in C^\infty(\mathscr D)$$. Then $$\phi\in C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$ vanishing on the boundary $$\partial \mathscr D$$ and $$-\phi''(x)\,{=}\,\psi(x)$$. We conclude from which (2.2) follows. The proof is complete. □ Next we define a sequence of approximations to the fractional noise $$\dot{W}$$. Let $$\{\mathscr D_i\,{=}\,(x_i,x_{i+1}],\ x_i\,{=}\,i h, i\,{=}\,0,1,\cdots,n-1\}$$, where $$h\,{=}\,1/n$$. We define the piecewise constant approximations of $$\dot{W}$$ by   W˙n(x)=∑i=0n−1χi(x)h∫DidW(y)n∈N, x∈D¯, (2.7) where $$\chi_i$$ is the characteristic function of $$\mathscr D_i$$. It is apparent that for each $$n\in \mathbb N$$, $$\dot{W}^n\in L^2(\mathscr D)$$ a.s. However, we have the following identity that shows that $$\mathbb E\left[\|\dot{W}^n\|^2\right]$$ is unbounded as $$h\rightarrow0$$:   E[‖W˙n‖2]=h2H−2∀ n∈N. (2.8) The following estimate will play an important role both in the proof of the existence of the weak solution of equation (1.2) and in the error estimate of piecewise constant approximations. Lemma 2.5 There exists $$C\,{=}\,C(H)$$ such that   ∑i≠jn∫Di∫Dj|x−y|2H−2dxdy≤Ch2H−1. (2.9) Proof. By direct calculation, for $$i,j\in \{0,1,\cdots,n-1\}$$ and $$i\neq j$$,   ∫Di∫Dj|x−y|2H−2dxdy=Ai,j(H)h2H2H(1−2H), where $$A_{i,j}(H)\,{=}\,2|i-j|^{2H}-|i-j+1|^{2H}-|i-j-1|^{2H}$$. A simple calculation implies that $$\sum\limits_{i\neq j}A_{i,j}(H)\,{=}\,(n-n^{2H})/2$$. As a consequence,   ∑i≠j∫Di∫Dj|x−y|2H−2dxdy=h2H2H(1−2H)∑i≠jAi,j(H)=h2H(n−n2H)H(1−2H)≤h2H−1H(1−2H), which proves (2.9) with $$C\,{=}\,\frac{1}{H(1-2H)}$$. □ Define the error between the two stochastic convolutions by $$E^n$$:   En(x):=∫DG(x,y)dW(y)−∫DG(x,y)dWn(y)x∈D. (2.10) From (2.7) we have   ∫DG(x,y)dWn(y)=∫D(∑i=0n−1χi(y)h∫DiG(x,z)dz)dW(y). Then we can rewrite $$E^n$$ as   En(x)=1h∑i=0n−1∫Di∫Di(G(x,y)−G(x,z))dzdW(y). Next we use Lemma 2.5 to derive an estimate for $$E^n$$. Proposition 2.6 There exists a constant $$C\,{=}\,C(H)$$ such that   supx∈DE[|En(x)|2]≤Ch2H+1. (2.11) Proof. Appyling Itô isometry formula (2.6), we obtain   E[|En(x)|2] =H(1−2H)2∫D∫D|[G(x,y)−G^(x,y)]−[G(x,z)−G^(x,z)]|2|y−z|2−2Hdydz +H∫D|G(x,y)−G^(x,y)|2(y2H−1+(1−y)2H−1)dy=:H(1−2H)2⋅I1+H⋅I2. (2.12) For $$I_1$$, we first split it into two parts as follows:   I1 =1h2∑i≠jn−1∫Di∫Di|∫DiG(x,u)−G(x,y)du−∫DjG(x,v)−G(x,z)dv|2|y−z|2−2Hdydz +∑i=0n−1∫Di∫Di|G(x,y)−G(x,z)|2|y−z|2−2Hdydz=:I11+I12. (2.13) Applying Hölder’s inequality and the estimate (2.9) in Lemma 2.5, we get   I11 ≤1h2∑i≠jn−1∫Di∫Dj∫Di∫Dj|[G(x,u)−G(x,y)]−[G(x,v)−G(x,z)]|2|y−z|2−2Hdudvdydz ≤2h2∑i≠jn−1∫Di∫Dj∫Di∫Dj|u−y|2+|v−z|2|y−z|2−2Hdudvdydz ≤4h2∑i≠jn−1∫Di∫Dj|y−z|2H−2dudvdydz≤4h2H+1. (2.14) Since the Green’s function is Lipschitz continuous,   I12≤∑i=0n−1∫Di∫Di|G(x,y)−G(x,z)|2|y−z|2−2Hdydz≤∑i=0n−1∫Di∫Di|y−z|dydz=2h2H+1(2H+1)(2H+2). (2.15) Next we evaluate $$I_2$$. Since the Green’s function $$G$$ is Lipschitz continuous,   I2=∑i=0n−1∫Di|1h∫DiG(x,u)−G(x,y)du|2(y2H−1+(1−y)2H−1)dy≤Ch2. (2.16) Combining (2.12)–(2.16), we obtain the desired estimate (2.11). □ For $$\phi\in L^2(\mathscr D)$$, define $$K\phi\,{:=}\,\int_{\mathscr D} G(\cdot,y)\phi(y)\,{\rm d}y$$. We also denote $$K\dot{W}\,{:=}\,\int_{\mathscr D} G(\cdot,y)\,{\rm d}W(y)$$. Set $$f(u)\,{=}\,f(\cdot,u(\cdot))$$. Then (2.2) can be rewritten as   u+Kf(u)=Kg+KW˙. (2.17) To prove the existence of a unique solution of Equation (2.17), we need the following inequality which can be derived from the Poincaré’s inequality (2.4) (see, e.g., Buckdahn & Pardoux, 1990, Lemma 2.4):   (Kϕ,ϕ)≥γ‖Kϕ‖2∀ ϕ∈L2(D). (2.18) Theorem 2.7 Let Assumption 2.3 hold. equation (1.2) possesses a unique mild solution. Proof. We first prove the uniqueness. Suppose that $$u$$ and $$v$$ solve equation (2.2). Then   u−v+K(f(u)−f(v))=0. Multiplying by $$f(u)-f(v)$$ on the above equation, we have   (u−v,f(u)−f(v))+(K(f(u)−f(v)),f(u)−f(v))=0. From the monotone-type condition (2.3) in Assumption 2.3 and (2.18), we deduce that   (γ−L)‖u−v‖2≤0, which implies that $$u\,{=}\,v$$. Next we prove the existence. The proof is for bounded $$f$$. The general case of $$f$$ satisfying the linear growth condition (2.5) follows from localization arguments in Buckdahn & Pardoux (1990, Theorem 2.5). For each $$n\in \mathbb N_+$$, we consider the SPDE obtained by replacing $$\dot W$$ with $$\dot W^n$$ in equation (1.2):   −d2dx2un+f(un) =g+W˙ninD,un =0 on∂D. (2.19) The existence of a unique solution $$u^n\in H^1_0(\mathscr D)$$ for equation (2.19) follows from the classical deterministic analysis. Clearly, $$u^n-u^m+K(\,f(u^n)-f(u^m))\,{=}\,K(\dot{W}^n-\dot{W}^m)$$. Multiplying by $$f(u^n)-f(u^m)$$, we obtain   (un−um,f(un)−f(um))+(K(f(un)−f(um)),f(un)−f(um))=(K(W˙n−W˙m),f(un)−f(um)). It follows from the monotone-type condition (2.3) and Poincarée inequality (2.18) that   (γ−L)‖un−um‖2≤(K(W˙n−W˙m),f(un)−f(um)+2γ(un−um)). (2.20) Since $$\mathbb E\left[\|K(\dot{W}^n-\dot{W}^m)\|^2\right]$$ tends to $$0$$ as $$n,m\rightarrow\infty$$ and $$f$$ is bounded, $$\{u^n\}$$ is a Cauchy sequence in $$L^2(\mathscr D\times \it{\Omega})$$. Hence, there exists $$u$$ in $$L^2(\mathscr D\times \it{\Omega})$$ such that $$u\,{=}\,\lim_{n\rightarrow\infty}u^n$$. From the boundedness of $$f$$ and Assumption 2.3, $$f(u^n)\rightarrow f(u)$$ in $$L^2(\mathscr D\times \it{\Omega})$$ as $$n\rightarrow\infty$$. The existence then follows from taking the limit in (2.19). □ 3. Error estimates of piecewise constant approximations In this section, we estimate the error between the solution of equation (1.2) and the solution of the approximate equation   −d2dx2un+f(un) =g+W˙ninD,un =0 on∂D. (3.1) Set $$F^n\,{=}\,g+\dot{W}^n$$. The variational formulation of equation (3.1) is to find a $$u^n\in H^1_0(\mathscr D)$$ such that a.s.   (ddxun,ddxv)+(f(un),v)=(Fn,v)∀ v∈H01(D). (3.2) We first analyse the regularity and obtain a bound for $$u^n$$, which will play a key role in the error estimate of the finite element approximation for equation (3.1) in Section 4. Theorem 3.1 Let Assumption 2.3 hold. Equation (3.2), therefore, equation (3.1), has a unique solution $$u^n\in H^1_0(\mathscr D)\cap H^2(\mathscr D)$$, a.s. Moreover, there exists a constant $$C$$ such that   E[‖un‖22]≤Ch2H−2. (3.3) Proof. The existence of a unique solution $$u^n\in H^1_0(\mathscr D)$$ a.s. follows from the classical deterministic arguments. To obtain (3.3), we first notice that Assumption 2.3, the Poincaré’s inequality (2.4) and Cauchy–Schwarz inequality yield that   ‖Fn‖⋅‖un‖≥(Fn,un)=‖ddxun‖2+(f(un),un)≥(γ−L)‖un‖2, from which we obtain   ‖un‖≤1γ−L‖Fn‖. Set $$R^n\,{=}\,F^n-f(u^n)$$. The linear growth condition (2.5) gives   ‖Rn‖2≤4β2+(2+4β2(γ−L)2)‖Fn‖2. On the other hand, it follows from equation (3.1) that $$u^n\in H^2(\mathscr D)$$ and   ‖un‖22≤C‖Rn‖2 for some $$C\in (0,\infty)$$. We conclude (3.3) by combining the above estimates and (2.8). □ Next we estimate the error between the exact solution $$u$$ of equation (1.2) and its approximation $$u^n$$ defined by equation (3.1). Recall that it follows from Definition 2.2 that $$u$$ and $$u^n$$ are the unique solutions of the following Hammerstein integral equations, respectively:   u+Kf(u)=Kg+KW˙, (3.4)  un+Kf(un)=Kg+KWn˙. (3.5) Theorem 3.2 Let Assumption 2.3 hold. There exists a constant $$C$$ such that   E[‖u−un‖2]≤ChH2+14. (3.6) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖u−un‖2]≤ChH+12. (3.7) Proof. Subtracting (3.5) from (3.4), we obtain   u(x)−un(x)+K(f(u)−f(un))=En. (3.8) In terms of the estimate (2.11) of $$E^n$$ defined by (2.10) in Lemma 2.6, to prove (3.6), it suffices to prove   ‖u−un‖2≤C‖En‖2+‖En‖. (3.9) To this end, we multiply (3.8) by $$f(u)-f(u^n)$$ to obtain   (u−un,f(u)−f(un))+(K(f(u)−f(un)),f(u)−f(un))=(En,f(u)−f(un)). The estimate (2.18) and the monotone-type condition (2.3) yield   −L‖u−un‖2+γ‖K(f(u)−f(un))‖2≤‖En‖⋅‖f(u)−f(un)‖. (3.10) Using the Young-type inequality $$\|\phi+\psi\|^2\,{\ge}\,\epsilon\|\phi\|^2-\frac{2-\epsilon}{1-\epsilon}\|\psi\|^2$$ with $$\phi\,{=}\,u-u^n,\psi\,{=}\,-E^n$$ and $$\epsilon\,{=}\,\frac{L+\gamma}{2\gamma}$$, we obtain   ‖K(f(u)−f(un))‖2=‖u−un−En‖2≥L+γ2γ‖u−un‖2−3γ−Lγ−L‖En‖2. (3.11) By the average inequality $$a b\,{\le}\, \frac{L-\gamma}{4\beta} a^2+\frac{\beta}{L-\gamma}b^2$$ and (2.5), we obtain   ‖En‖⋅‖f(u)−f(un)‖≤β‖En‖(1+‖u−un‖)≤β‖En‖+L−γ4‖u−un‖2+β2L−γ‖En‖2. (3.12) Substituting (3.12) and (3.11) into (3.10), we deduce that   −L‖u−un‖2+L+γ2‖u−un‖2−2(3γ−L)γ−L‖En‖2≤β‖En‖+L−γ4‖u−un‖2+β2L−γ‖En‖2, from which the desired estimate (3.9) follows. Now assume that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then the term $$\|E^n\|$$ in (3.9) would disappear. In this case, we achieve (3.7). □ 4. Finite element approximations In this section, we consider the finite element approximations of equation (3.2) and establish an overall error estimate between the exact solution and its finite element approximations. Let $$V_h$$ be the continuous piecewise linear finite element subspace of $$H^1_0(\mathscr D)$$ with respect to the quasi-uniform partition $$\{\mathscr D_i\}_{i=0}^{n-1}$$ given in Section 2. Then the finite element approximation of equation (3.2) is to find an $$u^n_h\in V_h$$ for each $$n\in \mathbb N$$ such that   (ddxuhn,ddxvh)+(f(uhn),vh)=(Fn,vh)∀ vh∈Vh. (4.1) Theorem 4.1 Let Assumption 2.3 hold. Equation (4.1) has a unique solution $$u^n_h\in H^1_0(\mathscr D)$$, a.s. Moreover, there exists a constant $$C$$ such that   E[‖uhn‖12]≤Ch2H−2. (4.2) Proof. Following a similar argument as in the proof of Theorem 3.1, we have   ‖uhn‖≤‖Fn‖γ−L. (4.3) Define $$R^n_h\,{=}\,F^n-f(u^n_h)$$. The linear growth condition (2.5) together with (4.3) implies   r−h‖Rhn‖2≤4β2+(2+4β2(γ−L)2)‖Fn‖2. (4.4) Notice that $$u^n_h$$ is the solution of   (ddxuhn,ddxvh)=(Rhn,vh)∀ vh∈Vh, from which we derive   ‖uhn‖12≤C‖Rhn‖2. (4.5) We conclude the estimate (4.2) with (4.3)–(4.5) and (2.8). □ Next we derive an estimate between $$u^n$$ and $$u^n_h$$. For this purpose, we introduce the Galerkin (or Ritz) projection operator $$\mathscr R_h: H^1_0(\mathscr D)\rightarrow V_h$$ defined by   (ddxRhw,ddxvh)=(ddxw,ddxvh)∀ vh∈Vh, w∈H01(D). (4.6) It is well known that there exists a constant $$C$$ such that (see, e.g., Thomée, 2006, Lemma 1.1)   ‖w−Rhw‖+h‖ddx(w−Rhw)‖≤Ch2‖w‖2∀ w∈H01(D)∩H2(D). (4.7) Theorem 4.2 Let Assumption 2.3 hold. There exists a constant $$C$$ such that   E[‖un−uhn‖2]≤ChH+12. (4.8) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖un−uhn‖2]≤ChH+1. (4.9) Proof. It follows from (3.2), (4.1) and (4.6) that   ‖ddx(Rhun−uhn)‖2+(f(un)−f(uhn),Rhun−uhn)=0. (4.10) The Assumptions 2.3 and the average inequality $$a\cdot b\,{\le}\, \frac{\gamma-L}{2\beta} a^2+\frac{\beta}{2(\gamma-L)}b^2$$ with $$a\,{=}\,\|u^n-u^n_h\|$$ and $$b\,{=}\,\|u^n-\mathscr R_hu^n\|$$ yield   ‖ddx(Rhun−uhn)‖2≤γ+L2‖un−uhn‖2+β‖un−Rhun‖+β22(γ−L)‖un−Rhun‖2. (4.11) Applying the projection theorem, Poincaré inequality (2.4) and the above inequality, we obtain   γ‖un−uhn‖2≤γ+L2‖un−uhn‖2+β‖un−Rhun‖+γ+β22(γ−L)‖un−Rhun‖2 (4.12) from which and (4.7) we derive   ‖un−uhn‖2≤C(‖un−Rhun‖+‖un−Rhun‖2)≤Ch2‖un‖2. (4.13) The desired error estimate then follows from (4.13) and Theorem 3.1. Now assume that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then the term $$\|u^n-\mathscr R_hu^n\|$$ in (4.11) would disappear. As a consequence,   ‖un−uhn‖2≤C‖un−Rhun‖2≤Ch4‖un‖22≤Ch2H+2. (4.14) This leads to the estimate (4.9). □ Remark 4.3 We should not expect any estimate of $$\mathbb E\left[\|\frac{\,{\rm d}}{\,{\rm d}x}(u^n-u^n_h)\|^2\right]$$ with a positive order since $$\mathbb E\left[\|u^n\|_2^2\right]\,{=}\,\mathscr O(h^{2H-2})$$. However, by the proof of Theorem 4.2,   E[‖ddx(Rhun−uhn)‖2]≤ChH+1, which agrees with the property of super-convergence of finite element method. Combining Theorem 3.2 and Theorem 4.2, we derive the main result about the error estimate between the exact solution $$u$$ and finite element solution $$u^n_h$$ by the triangle inequality. Theorem 4.4 Under Assumption 2.3, the error between the exact solution $$u$$ of equation (1.2) and its finite element solution $$u^n_h$$ defined by (4.1) satisfies   E[‖u−uhn‖2]≤ChH2+14. (4.15) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖u−uhn‖2]≤ChH+12. (4.16) Proof. The estimates (4.15) and (4.16) follows from (3.6), (3.7) in Theorem 3.2 and (4.8), (4.9) in Theorem 4.2. □ 5. Conclusions In this article, we developed the Galerkin finite element method for the boundary value problem of a one-dimensional second-order SDE driven by an fBm. The Hurst index $$H$$ of the fBm is assumed to be $$\,{\le}\,1/2$$. We proved that, with continuous piecewise linear finite elements, the mean square convergence rate of the finite element approximations in the case of Lipschitz coefficient is $$\mathscr O(h^{H+1/2})$$, which is consistent with the existing result for white noise (see, e.g. Allen et al., 1998; Gyöngy & Martínez, 2006). In a separate work (see Cao et al., 2016), we have obtained strong convergence rate of finite element approximations for one-dimensional time-dependent SPDEs, including nonlinear stochastic heat equation and stochastic wave equation, driven by a fractional Brownian sheet which is temporally white and spatially fractional with $$H\,{\le}\, 1/2$$. In future work, we plan to study the optimal convergence order of finite element approximations for SPDEs (1.2) in high-dimensional domains driven by a fractional Brownian sheet with $$H\,{\le}\, 1/2$$. Acknowlegdements The authors would like to thank the anonymous referee and the editor for their valuable comments on the first version of this paper. Funding Natural Science Foundation of China (No. 91530118, No. 91130003, No. 11021101 and No. 11290142). References Adams, R. A. & Fournier, J. F. ( 2003) Sobolev Spaces. Pure and Applied Mathematics (Amsterdam) , vol. 140, 2nd edn. Amsterdam: Elsevier/Academic Press, pp. xiv+ 305. Allen, E. J., Novosel, S. J. & Zhang, Z. ( 1998) Finite element and difference approximation of some linear stochastic partial differential equations. Stochastics Stochastic Rep.,  64, 117– 142. Google Scholar CrossRef Search ADS   Bardina, X. & Jolis, M. ( 2006) Multiple fractional integral with Hurst parameter less than $$\frac 12$$. Stochastic Process. Appl. , 116, 463– 479. Google Scholar CrossRef Search ADS   Buckdahn, R. & Pardoux, É. ( 1990) Monotonicity methods for white noise driven quasi-linear SPDEs. Diffusion Processes and Related Problems in Analysis , vol. I (Evanston, IL, 1989). Progr. Probab., 22. Boston, MA: Birkhäuser, pp. 219– 233. Google Scholar CrossRef Search ADS   Cao, Y., Yang, H. & Yin, L. ( 2007) Finite element methods for semilinear elliptic stochastic partial differential equations. Numer. Math. , 106, 181– 198. Google Scholar CrossRef Search ADS   Cao, Y., Hong, J. & Liu, Z. ( 2015) Well-posedness and finite element approximations for elliptic SPDEs with Gaussian noises. arXiv:1510.01873v4. Cao, Y., Hong, J. & Liu, Z. ( 2016) Approximating stochastic evolution equations with additive white and rough noises. arXiv:1601.02085. Du, Q. & Zhang, T. ( 2002) Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. , 40, 1421– 1445 (electronic). Google Scholar CrossRef Search ADS   Gyöngy, I. & Martínez, T. ( 2006) On numerical solution of stochastic partial differential equations of elliptic type. Stochastics , 78, 213– 231. Google Scholar CrossRef Search ADS   Hairer, M., Stuart, A. & Voss, J. ( 2011) Sampling conditioned hypoelliptic diffusions. Ann. Appl. Probab. , 21, 669– 698. Google Scholar CrossRef Search ADS   Khoshnevisan, D. ( 2009) A primer on stochastic partial differential equations. A Minicourse on Stochastic Partial Differential Equations . Lecture Notes in Math., vol. 1962. Berlin: Springer, pp. 1– 38. Google Scholar CrossRef Search ADS   Mandelbrot, B. B. & Van Ness, J. W. ( 1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev. , 10, 422– 437. Google Scholar CrossRef Search ADS   Martínez, T. & Sanz-Solé, M. ( 2006) A lattice scheme for stochastic partial differential equations of elliptic type in dimension $$d\geq 4$$. Appl. Math. Optim. , 54, 343– 368. Google Scholar CrossRef Search ADS   Shiryaev, A. N. ( 1999) Essentials of Stochastic Finance . Advanced Series on Statistical Science & Applied Probability, vol. 3. River Edge, NJ: World Scientific Publishing Co., Inc., pp. xvi+ 834. Google Scholar CrossRef Search ADS   Simonsen, I. ( 2003) Measuring anti-correlations in the nordic electricity spot market by wavelets. Phys. A , 322, 597– 606. Google Scholar CrossRef Search ADS   Thomée, V. ( 2006) Galerkin finite element methods for parabolic problems.  Springer Series in Computational Mathematics, vol. 25, second edn. Berlin: Springer, pp. xii+ 370. van Wyk, H.-W., Gunzburger, M., Burkhardt, J. & Stoyanov, M. ( 2015) Power-law noises over general spatial domains and on nonstandard meshes. SIAM/ASA J. Uncertain. Quantif. , 3, 296– 319. Google Scholar CrossRef Search ADS   Zhang, Z., Tretyakov, M. V., Rozovskii, B. & Karniadakis, G. E. ( 2015) Wiener chaos versus stochastic collocation methods for linear advection-diffusion-reaction equations with multiplicative white noise. SIAM J. Numer. Anal. , 53, 153– 183. Google Scholar CrossRef Search ADS   © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion

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Abstract

Abstract We consider finite element approximations for a one-dimensional second-order stochastic differential equation of boundary value type driven by a fractional Brownian motion with Hurst index $$H\,{\le}\, 1/2$$. We make use of a sequence of approximate solutions with the fractional noise replaced by its piecewise constant approximations to construct the finite element approximations for the equation. The error estimate of the approximations is derived through rigorous convergence analysis. 1 Introduction Many physical and engineering phenomena can be modeled by stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) when including some levels of uncertainties. The advantage of modeling with stochastic equations is that they are able to more fully capture the practical behavior of underlying models; it also means that the corresponding numerical analysis will require new tools to simulate the systems, produce the solutions and analyse the information stored within the solutions. Stochastic equations derived from fluid flows and other engineering fields are often driven by white noise. The white noise is an uncorrelated noise with delta function as its covariance. However, random fluctuations in complex systems may not be uncorrelated, i.e., they may not be white noise. Such noises are named as colored noises. As an important class of colored noises, the fractional Brownian motion (fBm)-type noise appears naturally in the modeling of many physical and social phenomena (see, e.g., Mandelbrot & Van Ness, 1968; van Wyk et al., 2015). For example, fBm is suitable in describing the widths of consecutive annual rings of a tree and the temperature at a specific place (see, e.g., Shiryaev, 1999); it can also be applied to simulate the turbulence in an incompressible fluid flow and the prices of electricity in a liberated electricity market (see, eg., Simonsen, 2003). As a centered Gaussian process, the fBm can be defined as follows. Let $$\mathscr D\,{=}\,(0,1)$$ and denote $${\overline {\mathscr D}}$$, $$\partial \mathscr D$$ the closure and the boundary of $$\mathscr D$$, respectively. The fBm $$W\,{=}\,\{W(x),\ x\in {\overline {\mathscr D}}\}$$ on $${\overline {\mathscr D}}$$ is determined by its covariance function   Cov(x,y):=E[W(x)W(y)]=x2H+y2H−|x−y|2H2x,y∈D¯. (1.1) Here, $$H\in (0,1)$$ is the so-called Hurst index. The fBms with $$H<1/2$$ and $$H>1/2$$ are significantly different both physically and mathematically. In the first two aforementioned applications, the corresponding fBms have Hurst index $$H>1/2$$. In such cases, the physical process presents an aggregation and persistent behavior. On the other hand, the Hurst index $$H$$ is less than $$1/2$$ in the last two cases, where the process is anti-persistent and may have long-range negative interactions. These two classes of fBms are separated by the standard Brownian motion whose Hurst index is $$H\,{=}\,1/2$$. Mathematically, the fBm with $$H>1/2$$ is a Gaussian process whose covariance function has a bounded variation on $${\overline {\mathscr D}}\times {\overline {\mathscr D}}$$. The stochastic integral against the fBm with $$H>1/2$$ can be viewed as a pathwise Riemann–Stieltjes integral (or Young integral) and classical methods are applicative. On the contrary, the covariance function of the fBm with $$H\,{\le}\, 1/2$$ does not have bounded variation. This posts a particular difficulty when studying SDEs or SPDEs driven by such noises. The main objective of this article is to investigate the well posedness and finite element approximations for the following second-order SDE of boundary type driven by an fBm with $$H\,{\le}\, 1/2$$:   −d2dx2u(x)+f(x,u(x)) =g(x)+W˙(x)x∈D,u(x) =0,x∈∂D. (1.2) Here, $$g: \mathscr D\rightarrow \mathbb R$$ is square integrable, $$f: \mathscr D\times\mathbb R \rightarrow \mathbb R$$ satisfies certain conditions given in Section 2 and $$W\,{=}\,\{W(x):\ x\in {\overline {\mathscr D}}\}$$ is an fBm, determined by (1.1) with $$H\,{\le}\, 1/2$$, on a filtered probability space $$(\it{\Omega}, \mathscr F, (\mathscr F_x)_{x\in {\overline {\mathscr D}}}, \mathbb P)$$. The homogenous Dirichlet boundary condition in equation (1.2) corresponds to a second-order SDE conditioned to hit a particular point at ‘time’ $$x\,{=}\,1$$. As such it is a generalization to general fractional noise of the conditioned diffusions studied in Hairer et al. (2011). On the other hand, this equation can be considered as an elliptic SPDE in one dimension. Note that one can also study homogenous Neumann boundary condition and the main results of this article are also valid. Equation (1.2) driven by the white noise, i.e., $$H\,{=}\,1/2$$, has been considered by several authors (see, e.g., Allen et al., 1998; Du & Zhang, 2002; Gyöngy & Martínez, 2006; Martínez & Sanz-Solé, 2006; Cao et al., 2007, 2015; Zhang et al., 2015). Allen et al. (1998) investigated the finite difference and finite element approximations of the linear case of equation (1.2). They proved the first-order convergence for both the finite difference and finite element approximations. The three authors in Cao et al. (2015) investigated the finite element approximations of equation (1.2) in possibly any dimensions formulated in the form of Karhunen–Loève expansions for certain Gaussian noises. For the case where $$H>1/2$$, the well posedness and finite difference approximations can be studied using the methodology of Martínez & Sanz-Solé (2006); by treating the fBm as a colored noise with a special Riesz kernel. However, the method of treating the white noise and more regular noises does not apply to fractional noise with $$H<1/2$$, since the exact solution is less regular. To the best of our knowledge, there have not been literatures studying numerical approximations for SDEs or SPDEs driven by fractional noises with $$H<1/2$$. The primary challenge in studying the finite element approximations of equation (1.2) driven by fBm with $$H<1/2$$ is threefold: (i) as a colored noise, the increments of the fBm in two disjoint intervals are not independent; (ii) the regularity of $$\dot{W}$$ with $$H<1/2$$ is very low, and (iii) the approach of Karhunen–Loève expansions used in Cao et al. (2015) fails. In this article, we study the well posedness and the finite element approximations of equation (1.2) through a special Itô isometry which is only valid for $$H\,{\le}\, 1/2$$ (see, (2.6)). Using this isometry, we obtain the existence of a unique solution for equation (1.2) by analysing the convergence of a sequence of approximate solutions of SPDEs with the fractional noise replaced by a sum of tensor products between correlated Gaussian random variables and piecewise constant functions in the physical domain. Following the well-posedness analysis, we construct the finite element approximations of equation (1.2) through two steps. In the first step, we derive an error estimate between the exact solution and its approximations that are used in the well-posedness analysis. This error estimate also heavily depends on the aforementioned Itô isometry. In the second step, we apply the Galerkin finite element method to the approximate noise-driven SPDE, and obtain the overall error estimate of the finite element solution through an finite element error estimate for the approximate SPDE. The article is organized as follows. First, we define the weak solution and mild solution of equation (1.2) and establish their existence and uniqueness in Section 2. Next in Section 3, we derive the error estimate between the exact solution of equation (1.2) and the solution of the approximate SPDE. In Section 4, we apply a finite element method to this approximate SPDE and derive the overall error estimate of the finite element solution. Finally, a few concluding remarks are given in Section 5. We conclude this section by introducing some notations that will be used throughout the article. Denote by $$L^2(\mathscr D)$$ the space of square integrable functions in $$\mathscr D$$ with its inner product and norm denoted by $$(\cdot,\cdot)$$ and $$\|\cdot\|$$, respectively. For $$r>0$$, we use $$H^r(\mathscr D)$$ to denote the usual Sobolev space whose norm is denoted by $$\|\cdot\|_r$$. We also use $$H^1_0(\mathscr D)$$ to denote the subspace of $$H^1(\mathscr D)$$ whose elements vanish on $$\partial \mathscr D$$. We denote by $$C$$ a generic positive constant independent of either the truncation number $$n$$ or the grid size $$h$$, which will change from one line to another. 2. Well posedness of the problem In this section, we define the weak solution and mild solution of equation (1.2), and then establish their equivalence, existence and uniqueness. Definition 2.1 An $$\mathscr F_x$$-adapted stochastic process $$u\,{=}\,\{u(x):x\in \mathscr D\}$$ is called a weak solution of equation (1.2), if for every $$\phi\in C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$ vanishing on $$\partial \mathscr D$$ it holds a.s. that   −∫Du(x)ϕ″(x)dx+∫Df(x,u(x))ϕ(x)dx=∫Dg(x)ϕ(x)dx+∫Dϕ(x)dW(x). (2.1) Definition 2.2 An $$\mathscr F_x$$-adapted stochastic process $$u\,{=}\,\{u(x):x\in \mathscr D\}$$ is called a mild solution of equation (1.2), if for all $$x\in \mathscr D$$ it holds a.s. that   u(x)+∫DG(x,y)f(y,u(y))dy=∫DG(x,y)g(y)dy+∫DG(x,y)dW(y), (2.2) where $$G$$ is the Green’s function associated with the Poisson equation with Dirichlet boundary. It is well known that the related Green’s function $$G$$ is given by $$G(x,y)\,{=}\,x\wedge y-xy$$$$x,y\in {\overline {\mathscr D}}$$. Obviously, $$G$$ is Lipschitz continuous over $${\overline {\mathscr D}}\times {\overline {\mathscr D}}$$. Without loss of generality, we assume that $$f(x,0)\,{=}\,0$$ for any $$x\in \mathscr D$$. Otherwise, we simply replace $$f(x,r)$$ by $$f(x,r)-f(x,0)$$ and $$g(x)$$ by $$g(x)-f(x,0)$$. Assume furthermore that $$f$$ satisfies the following assumptions. Assumption 2.3 (1) (Monotone-type condition) There exists a positive constant $$L<\gamma$$ such that   (f(x,r)−f(x,s),r−s)≥−L|r−s|2∀ x∈D, r,s∈R, (2.3) where $$\gamma$$ is the positive constant in the Poincaré inequality (see, e.g., Adams & Fournier, 2003, Theorem 6.30):   ‖ddxv‖2≥γ‖v‖2∀ v∈H01(D). (2.4) (2) (Linear growth condition) There exists a positive constant $$\beta$$ such that   |f(x,r)−f(x,s)|≤β(1+|r−s|)∀ x∈D, r,s∈R. (2.5) We remark that these two conditions can be satisfied when $$f$$ is a sum of a nondecreasing bounded function and a Lipschitz continuous function with the Lipschitz constant less than $$\gamma$$ (see, e.g., Buckdahn & Pardoux, 1990; Gyöngy & Martínez, 2006). In the case, $$\mathscr D\,{=}\,(0,1),$$ it can be easily shown that $$\gamma \,{=}\,2$$. Therefore, we assume that $$L<2$$ throughout the rest of this article. Before establishing the well posedness of equation (1.2), we follow the approach of Bardina & Jolis (2006) to define stochastic integral with respect to the fBm $$W$$ with $$H<1/2$$. To this end, we introduce the set $$\Phi$$ of all step functions on $$\mathscr D$$ of the form   f=∑j=0N−1fjχ(aj,aj+1], where $$0\,{=}\,a_0<a_1<\cdots<a_N\,{=}\,1$$ is a partition of $$\mathscr D$$ and $$f_j\in\mathbb R$$, $$j\,{=}\,0,1,\cdots,N-1$$, $$N\in \mathbb N_+$$. For $$f\in \it{\Phi}$$, we define its integral with respect to $$W$$ by Riemann sum as   I(f)=∑j=0N−1fj(W(aj+1)−W(aj)), and for $$f,g\in \it{\Phi}$$, we define their scalar product as   Ψ(f,g):=E[I(f)I(g)]. Next we extend $$\it{\Phi}$$ through completion to a Hilbert space, denoted by $$\it{\Phi}^H$$. By Bardina & Jolis (2006, Lemma 2.1), we have a characterization of $$\it{\Phi}^H$$ through Itô isometry for simple functions:   Ψ(f,g) =H(1−2H)2∫D∫D(f(x)−f(y))(g(x)−g(y))|x−y|2−2Hdxdy +H∫Df(x)g(x)(x2H−1+(1−x)2H−1)dx∀ f,g∈Φ. (2.6) This shows that   ΦH={f∈L2(D¯):∫R∫R|f¯(x)−f¯(y)|2|x−y|2−2Hdxdy<∞}, where $$\overline{f}(x)\,{=}\,f(x)$$ when $$x\in{\overline {\mathscr D}}$$ and $$\overline{f}(x)\,{=}\,0$$ otherwise. As a consequence, the integral $$I$$ for a measurable deterministic function $$f:{\overline {\mathscr D}}\rightarrow \mathbb R$$ with respect to the fBm $$W$$ is an isometry between $$\it{\Phi}^H$$ and a subspace of $$L^2(\mathbb P)$$. Lemma 2.4 (1) The stochastic process $$\{v(x)\,{:=}\,\int_{\mathscr D} G(x,y)dW(y),\ x\in{\overline {\mathscr D}}\}$$ possesses an a.s. continuous modification. (2) Definitions 2.1 and 2.2 are equivalent to each other. Proof. Let $$x_1,x_2\in \mathscr D$$. The Ito’s isometry (2.6) yields   E[|v(x1)−v(x2)|2] =H(1−2H)2∫D∫D|[G(x1,y)−G(x2,y)]−[G(x1,z)−G(x2,z)]|2|y−z|2−2Hdydz +H∫D|G(x1,y)−G(x2,y)|2(y2H−1+(1−y)2H−1)dx. Since $$G\,{=}\,\{G(x,y):\ x,y\in {\overline {\mathscr D}}\}$$ is Lipschitz continuous with respect to both $$x$$ and $$y$$, we have   |[G(x1,y)−G(x2,y)]−[G(x1,z)−G(x2,z)]|2≤2|x1−x2|×2|y−z|. Direct calculations yield that   ∫D∫D|y−z|2H−1dydz=H(1+2H). Therefore, there exists $$C\,{=}\,C(H)$$ such that   E[|v(x1)−v(x2)|2]≤C|x1−x2|x1,x2∈D¯, from which, and the fact that $$v$$ is Gaussian, we conclude that $$v$$ has an a.s. continuous modification (see, e.g., Khoshnevisan, 2009, Exercise 4.9). Assume that $$u$$ satisfies (2.2) and let $$\phi\in C_0^\infty(\mathscr D)$$. Multiplying (2.2) by $$\phi''(x)$$, integrating over $$\mathscr D$$ and using the identity $$-\int_{\mathscr D} G(x,y) \phi''(y)\,{\rm d}y\,{=}\,\phi(x)$$, we obtain (2.1) for smooth $$\phi$$. The general case follows from the fact that $$C_0^\infty(\mathscr D)$$ is dense in $$C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$. Suppose now that $$u$$ satisfies (2.1). Choose $$\phi(x)\,{=}\,-\int_{\mathscr D} G(x,y)\psi(y)\,{\rm d}y$$ with $$\psi\in C^\infty(\mathscr D)$$. Then $$\phi\in C^2(\mathscr D)\cap C({\overline {\mathscr D}})$$ vanishing on the boundary $$\partial \mathscr D$$ and $$-\phi''(x)\,{=}\,\psi(x)$$. We conclude from which (2.2) follows. The proof is complete. □ Next we define a sequence of approximations to the fractional noise $$\dot{W}$$. Let $$\{\mathscr D_i\,{=}\,(x_i,x_{i+1}],\ x_i\,{=}\,i h, i\,{=}\,0,1,\cdots,n-1\}$$, where $$h\,{=}\,1/n$$. We define the piecewise constant approximations of $$\dot{W}$$ by   W˙n(x)=∑i=0n−1χi(x)h∫DidW(y)n∈N, x∈D¯, (2.7) where $$\chi_i$$ is the characteristic function of $$\mathscr D_i$$. It is apparent that for each $$n\in \mathbb N$$, $$\dot{W}^n\in L^2(\mathscr D)$$ a.s. However, we have the following identity that shows that $$\mathbb E\left[\|\dot{W}^n\|^2\right]$$ is unbounded as $$h\rightarrow0$$:   E[‖W˙n‖2]=h2H−2∀ n∈N. (2.8) The following estimate will play an important role both in the proof of the existence of the weak solution of equation (1.2) and in the error estimate of piecewise constant approximations. Lemma 2.5 There exists $$C\,{=}\,C(H)$$ such that   ∑i≠jn∫Di∫Dj|x−y|2H−2dxdy≤Ch2H−1. (2.9) Proof. By direct calculation, for $$i,j\in \{0,1,\cdots,n-1\}$$ and $$i\neq j$$,   ∫Di∫Dj|x−y|2H−2dxdy=Ai,j(H)h2H2H(1−2H), where $$A_{i,j}(H)\,{=}\,2|i-j|^{2H}-|i-j+1|^{2H}-|i-j-1|^{2H}$$. A simple calculation implies that $$\sum\limits_{i\neq j}A_{i,j}(H)\,{=}\,(n-n^{2H})/2$$. As a consequence,   ∑i≠j∫Di∫Dj|x−y|2H−2dxdy=h2H2H(1−2H)∑i≠jAi,j(H)=h2H(n−n2H)H(1−2H)≤h2H−1H(1−2H), which proves (2.9) with $$C\,{=}\,\frac{1}{H(1-2H)}$$. □ Define the error between the two stochastic convolutions by $$E^n$$:   En(x):=∫DG(x,y)dW(y)−∫DG(x,y)dWn(y)x∈D. (2.10) From (2.7) we have   ∫DG(x,y)dWn(y)=∫D(∑i=0n−1χi(y)h∫DiG(x,z)dz)dW(y). Then we can rewrite $$E^n$$ as   En(x)=1h∑i=0n−1∫Di∫Di(G(x,y)−G(x,z))dzdW(y). Next we use Lemma 2.5 to derive an estimate for $$E^n$$. Proposition 2.6 There exists a constant $$C\,{=}\,C(H)$$ such that   supx∈DE[|En(x)|2]≤Ch2H+1. (2.11) Proof. Appyling Itô isometry formula (2.6), we obtain   E[|En(x)|2] =H(1−2H)2∫D∫D|[G(x,y)−G^(x,y)]−[G(x,z)−G^(x,z)]|2|y−z|2−2Hdydz +H∫D|G(x,y)−G^(x,y)|2(y2H−1+(1−y)2H−1)dy=:H(1−2H)2⋅I1+H⋅I2. (2.12) For $$I_1$$, we first split it into two parts as follows:   I1 =1h2∑i≠jn−1∫Di∫Di|∫DiG(x,u)−G(x,y)du−∫DjG(x,v)−G(x,z)dv|2|y−z|2−2Hdydz +∑i=0n−1∫Di∫Di|G(x,y)−G(x,z)|2|y−z|2−2Hdydz=:I11+I12. (2.13) Applying Hölder’s inequality and the estimate (2.9) in Lemma 2.5, we get   I11 ≤1h2∑i≠jn−1∫Di∫Dj∫Di∫Dj|[G(x,u)−G(x,y)]−[G(x,v)−G(x,z)]|2|y−z|2−2Hdudvdydz ≤2h2∑i≠jn−1∫Di∫Dj∫Di∫Dj|u−y|2+|v−z|2|y−z|2−2Hdudvdydz ≤4h2∑i≠jn−1∫Di∫Dj|y−z|2H−2dudvdydz≤4h2H+1. (2.14) Since the Green’s function is Lipschitz continuous,   I12≤∑i=0n−1∫Di∫Di|G(x,y)−G(x,z)|2|y−z|2−2Hdydz≤∑i=0n−1∫Di∫Di|y−z|dydz=2h2H+1(2H+1)(2H+2). (2.15) Next we evaluate $$I_2$$. Since the Green’s function $$G$$ is Lipschitz continuous,   I2=∑i=0n−1∫Di|1h∫DiG(x,u)−G(x,y)du|2(y2H−1+(1−y)2H−1)dy≤Ch2. (2.16) Combining (2.12)–(2.16), we obtain the desired estimate (2.11). □ For $$\phi\in L^2(\mathscr D)$$, define $$K\phi\,{:=}\,\int_{\mathscr D} G(\cdot,y)\phi(y)\,{\rm d}y$$. We also denote $$K\dot{W}\,{:=}\,\int_{\mathscr D} G(\cdot,y)\,{\rm d}W(y)$$. Set $$f(u)\,{=}\,f(\cdot,u(\cdot))$$. Then (2.2) can be rewritten as   u+Kf(u)=Kg+KW˙. (2.17) To prove the existence of a unique solution of Equation (2.17), we need the following inequality which can be derived from the Poincaré’s inequality (2.4) (see, e.g., Buckdahn & Pardoux, 1990, Lemma 2.4):   (Kϕ,ϕ)≥γ‖Kϕ‖2∀ ϕ∈L2(D). (2.18) Theorem 2.7 Let Assumption 2.3 hold. equation (1.2) possesses a unique mild solution. Proof. We first prove the uniqueness. Suppose that $$u$$ and $$v$$ solve equation (2.2). Then   u−v+K(f(u)−f(v))=0. Multiplying by $$f(u)-f(v)$$ on the above equation, we have   (u−v,f(u)−f(v))+(K(f(u)−f(v)),f(u)−f(v))=0. From the monotone-type condition (2.3) in Assumption 2.3 and (2.18), we deduce that   (γ−L)‖u−v‖2≤0, which implies that $$u\,{=}\,v$$. Next we prove the existence. The proof is for bounded $$f$$. The general case of $$f$$ satisfying the linear growth condition (2.5) follows from localization arguments in Buckdahn & Pardoux (1990, Theorem 2.5). For each $$n\in \mathbb N_+$$, we consider the SPDE obtained by replacing $$\dot W$$ with $$\dot W^n$$ in equation (1.2):   −d2dx2un+f(un) =g+W˙ninD,un =0 on∂D. (2.19) The existence of a unique solution $$u^n\in H^1_0(\mathscr D)$$ for equation (2.19) follows from the classical deterministic analysis. Clearly, $$u^n-u^m+K(\,f(u^n)-f(u^m))\,{=}\,K(\dot{W}^n-\dot{W}^m)$$. Multiplying by $$f(u^n)-f(u^m)$$, we obtain   (un−um,f(un)−f(um))+(K(f(un)−f(um)),f(un)−f(um))=(K(W˙n−W˙m),f(un)−f(um)). It follows from the monotone-type condition (2.3) and Poincarée inequality (2.18) that   (γ−L)‖un−um‖2≤(K(W˙n−W˙m),f(un)−f(um)+2γ(un−um)). (2.20) Since $$\mathbb E\left[\|K(\dot{W}^n-\dot{W}^m)\|^2\right]$$ tends to $$0$$ as $$n,m\rightarrow\infty$$ and $$f$$ is bounded, $$\{u^n\}$$ is a Cauchy sequence in $$L^2(\mathscr D\times \it{\Omega})$$. Hence, there exists $$u$$ in $$L^2(\mathscr D\times \it{\Omega})$$ such that $$u\,{=}\,\lim_{n\rightarrow\infty}u^n$$. From the boundedness of $$f$$ and Assumption 2.3, $$f(u^n)\rightarrow f(u)$$ in $$L^2(\mathscr D\times \it{\Omega})$$ as $$n\rightarrow\infty$$. The existence then follows from taking the limit in (2.19). □ 3. Error estimates of piecewise constant approximations In this section, we estimate the error between the solution of equation (1.2) and the solution of the approximate equation   −d2dx2un+f(un) =g+W˙ninD,un =0 on∂D. (3.1) Set $$F^n\,{=}\,g+\dot{W}^n$$. The variational formulation of equation (3.1) is to find a $$u^n\in H^1_0(\mathscr D)$$ such that a.s.   (ddxun,ddxv)+(f(un),v)=(Fn,v)∀ v∈H01(D). (3.2) We first analyse the regularity and obtain a bound for $$u^n$$, which will play a key role in the error estimate of the finite element approximation for equation (3.1) in Section 4. Theorem 3.1 Let Assumption 2.3 hold. Equation (3.2), therefore, equation (3.1), has a unique solution $$u^n\in H^1_0(\mathscr D)\cap H^2(\mathscr D)$$, a.s. Moreover, there exists a constant $$C$$ such that   E[‖un‖22]≤Ch2H−2. (3.3) Proof. The existence of a unique solution $$u^n\in H^1_0(\mathscr D)$$ a.s. follows from the classical deterministic arguments. To obtain (3.3), we first notice that Assumption 2.3, the Poincaré’s inequality (2.4) and Cauchy–Schwarz inequality yield that   ‖Fn‖⋅‖un‖≥(Fn,un)=‖ddxun‖2+(f(un),un)≥(γ−L)‖un‖2, from which we obtain   ‖un‖≤1γ−L‖Fn‖. Set $$R^n\,{=}\,F^n-f(u^n)$$. The linear growth condition (2.5) gives   ‖Rn‖2≤4β2+(2+4β2(γ−L)2)‖Fn‖2. On the other hand, it follows from equation (3.1) that $$u^n\in H^2(\mathscr D)$$ and   ‖un‖22≤C‖Rn‖2 for some $$C\in (0,\infty)$$. We conclude (3.3) by combining the above estimates and (2.8). □ Next we estimate the error between the exact solution $$u$$ of equation (1.2) and its approximation $$u^n$$ defined by equation (3.1). Recall that it follows from Definition 2.2 that $$u$$ and $$u^n$$ are the unique solutions of the following Hammerstein integral equations, respectively:   u+Kf(u)=Kg+KW˙, (3.4)  un+Kf(un)=Kg+KWn˙. (3.5) Theorem 3.2 Let Assumption 2.3 hold. There exists a constant $$C$$ such that   E[‖u−un‖2]≤ChH2+14. (3.6) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖u−un‖2]≤ChH+12. (3.7) Proof. Subtracting (3.5) from (3.4), we obtain   u(x)−un(x)+K(f(u)−f(un))=En. (3.8) In terms of the estimate (2.11) of $$E^n$$ defined by (2.10) in Lemma 2.6, to prove (3.6), it suffices to prove   ‖u−un‖2≤C‖En‖2+‖En‖. (3.9) To this end, we multiply (3.8) by $$f(u)-f(u^n)$$ to obtain   (u−un,f(u)−f(un))+(K(f(u)−f(un)),f(u)−f(un))=(En,f(u)−f(un)). The estimate (2.18) and the monotone-type condition (2.3) yield   −L‖u−un‖2+γ‖K(f(u)−f(un))‖2≤‖En‖⋅‖f(u)−f(un)‖. (3.10) Using the Young-type inequality $$\|\phi+\psi\|^2\,{\ge}\,\epsilon\|\phi\|^2-\frac{2-\epsilon}{1-\epsilon}\|\psi\|^2$$ with $$\phi\,{=}\,u-u^n,\psi\,{=}\,-E^n$$ and $$\epsilon\,{=}\,\frac{L+\gamma}{2\gamma}$$, we obtain   ‖K(f(u)−f(un))‖2=‖u−un−En‖2≥L+γ2γ‖u−un‖2−3γ−Lγ−L‖En‖2. (3.11) By the average inequality $$a b\,{\le}\, \frac{L-\gamma}{4\beta} a^2+\frac{\beta}{L-\gamma}b^2$$ and (2.5), we obtain   ‖En‖⋅‖f(u)−f(un)‖≤β‖En‖(1+‖u−un‖)≤β‖En‖+L−γ4‖u−un‖2+β2L−γ‖En‖2. (3.12) Substituting (3.12) and (3.11) into (3.10), we deduce that   −L‖u−un‖2+L+γ2‖u−un‖2−2(3γ−L)γ−L‖En‖2≤β‖En‖+L−γ4‖u−un‖2+β2L−γ‖En‖2, from which the desired estimate (3.9) follows. Now assume that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then the term $$\|E^n\|$$ in (3.9) would disappear. In this case, we achieve (3.7). □ 4. Finite element approximations In this section, we consider the finite element approximations of equation (3.2) and establish an overall error estimate between the exact solution and its finite element approximations. Let $$V_h$$ be the continuous piecewise linear finite element subspace of $$H^1_0(\mathscr D)$$ with respect to the quasi-uniform partition $$\{\mathscr D_i\}_{i=0}^{n-1}$$ given in Section 2. Then the finite element approximation of equation (3.2) is to find an $$u^n_h\in V_h$$ for each $$n\in \mathbb N$$ such that   (ddxuhn,ddxvh)+(f(uhn),vh)=(Fn,vh)∀ vh∈Vh. (4.1) Theorem 4.1 Let Assumption 2.3 hold. Equation (4.1) has a unique solution $$u^n_h\in H^1_0(\mathscr D)$$, a.s. Moreover, there exists a constant $$C$$ such that   E[‖uhn‖12]≤Ch2H−2. (4.2) Proof. Following a similar argument as in the proof of Theorem 3.1, we have   ‖uhn‖≤‖Fn‖γ−L. (4.3) Define $$R^n_h\,{=}\,F^n-f(u^n_h)$$. The linear growth condition (2.5) together with (4.3) implies   r−h‖Rhn‖2≤4β2+(2+4β2(γ−L)2)‖Fn‖2. (4.4) Notice that $$u^n_h$$ is the solution of   (ddxuhn,ddxvh)=(Rhn,vh)∀ vh∈Vh, from which we derive   ‖uhn‖12≤C‖Rhn‖2. (4.5) We conclude the estimate (4.2) with (4.3)–(4.5) and (2.8). □ Next we derive an estimate between $$u^n$$ and $$u^n_h$$. For this purpose, we introduce the Galerkin (or Ritz) projection operator $$\mathscr R_h: H^1_0(\mathscr D)\rightarrow V_h$$ defined by   (ddxRhw,ddxvh)=(ddxw,ddxvh)∀ vh∈Vh, w∈H01(D). (4.6) It is well known that there exists a constant $$C$$ such that (see, e.g., Thomée, 2006, Lemma 1.1)   ‖w−Rhw‖+h‖ddx(w−Rhw)‖≤Ch2‖w‖2∀ w∈H01(D)∩H2(D). (4.7) Theorem 4.2 Let Assumption 2.3 hold. There exists a constant $$C$$ such that   E[‖un−uhn‖2]≤ChH+12. (4.8) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖un−uhn‖2]≤ChH+1. (4.9) Proof. It follows from (3.2), (4.1) and (4.6) that   ‖ddx(Rhun−uhn)‖2+(f(un)−f(uhn),Rhun−uhn)=0. (4.10) The Assumptions 2.3 and the average inequality $$a\cdot b\,{\le}\, \frac{\gamma-L}{2\beta} a^2+\frac{\beta}{2(\gamma-L)}b^2$$ with $$a\,{=}\,\|u^n-u^n_h\|$$ and $$b\,{=}\,\|u^n-\mathscr R_hu^n\|$$ yield   ‖ddx(Rhun−uhn)‖2≤γ+L2‖un−uhn‖2+β‖un−Rhun‖+β22(γ−L)‖un−Rhun‖2. (4.11) Applying the projection theorem, Poincaré inequality (2.4) and the above inequality, we obtain   γ‖un−uhn‖2≤γ+L2‖un−uhn‖2+β‖un−Rhun‖+γ+β22(γ−L)‖un−Rhun‖2 (4.12) from which and (4.7) we derive   ‖un−uhn‖2≤C(‖un−Rhun‖+‖un−Rhun‖2)≤Ch2‖un‖2. (4.13) The desired error estimate then follows from (4.13) and Theorem 3.1. Now assume that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then the term $$\|u^n-\mathscr R_hu^n\|$$ in (4.11) would disappear. As a consequence,   ‖un−uhn‖2≤C‖un−Rhun‖2≤Ch4‖un‖22≤Ch2H+2. (4.14) This leads to the estimate (4.9). □ Remark 4.3 We should not expect any estimate of $$\mathbb E\left[\|\frac{\,{\rm d}}{\,{\rm d}x}(u^n-u^n_h)\|^2\right]$$ with a positive order since $$\mathbb E\left[\|u^n\|_2^2\right]\,{=}\,\mathscr O(h^{2H-2})$$. However, by the proof of Theorem 4.2,   E[‖ddx(Rhun−uhn)‖2]≤ChH+1, which agrees with the property of super-convergence of finite element method. Combining Theorem 3.2 and Theorem 4.2, we derive the main result about the error estimate between the exact solution $$u$$ and finite element solution $$u^n_h$$ by the triangle inequality. Theorem 4.4 Under Assumption 2.3, the error between the exact solution $$u$$ of equation (1.2) and its finite element solution $$u^n_h$$ defined by (4.1) satisfies   E[‖u−uhn‖2]≤ChH2+14. (4.15) Assume furthermore that $$f$$ is Lipschitz continuous with the Lipschitz constant $$L<\gamma$$, then   E[‖u−uhn‖2]≤ChH+12. (4.16) Proof. The estimates (4.15) and (4.16) follows from (3.6), (3.7) in Theorem 3.2 and (4.8), (4.9) in Theorem 4.2. □ 5. Conclusions In this article, we developed the Galerkin finite element method for the boundary value problem of a one-dimensional second-order SDE driven by an fBm. The Hurst index $$H$$ of the fBm is assumed to be $$\,{\le}\,1/2$$. We proved that, with continuous piecewise linear finite elements, the mean square convergence rate of the finite element approximations in the case of Lipschitz coefficient is $$\mathscr O(h^{H+1/2})$$, which is consistent with the existing result for white noise (see, e.g. Allen et al., 1998; Gyöngy & Martínez, 2006). In a separate work (see Cao et al., 2016), we have obtained strong convergence rate of finite element approximations for one-dimensional time-dependent SPDEs, including nonlinear stochastic heat equation and stochastic wave equation, driven by a fractional Brownian sheet which is temporally white and spatially fractional with $$H\,{\le}\, 1/2$$. In future work, we plan to study the optimal convergence order of finite element approximations for SPDEs (1.2) in high-dimensional domains driven by a fractional Brownian sheet with $$H\,{\le}\, 1/2$$. 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Published: Jan 1, 2018

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