We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $$A(x) : D^2 u(x) = f(x)$$ A ( x ) : D 2 u ( x ) = f ( x ) in a bounded but not necessarily convex domain $$\Omega $$ Ω and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale $$\epsilon $$ ϵ is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle for any uniformly positive definite matrix A provided that the mesh is face weakly acute. We establish a discrete Alexandroff–Bakelman–Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to pointwise error estimates of the form $$\begin{aligned} \Vert \,u - u^{\epsilon }_h\,\Vert _{L^{\infty }(\Omega )} \le \, C(A,u) \, h^{2\alpha /(2 + \alpha )} \big | \ln h \big | \qquad 0< \alpha \le 2, \end{aligned}$$ ‖ u - u h ϵ ‖ L ∞ ( Ω ) ≤ C ( A , u ) h 2 α / ( 2 + α ) | ln h | 0 < α ≤ 2 , provided $$\epsilon \approx h^{2/(2+\alpha )}$$ ϵ ≈ h 2 / ( 2 + α ) . Such a convergence rate is at best of order $$ h \big | \ln h \big |$$ h | ln h | , which turns out to be quasi-optimal.
Foundations of Computational Mathematics – Springer Journals
Published: Mar 1, 2017
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