2014/11/21 by Ricardo H. Nochetto, Nochetto, Ricardo H., Wujun Zhang +1 · 3 citations
Engineering · Computer Science · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1411.6036
We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form A(x) : D2 u(x) = f(x) in a bounded but not necessarily convex domain Ω and study it in the max norm. The fine scale is given by the meshsize h whereas the coarse scale ε is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) 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 ‖ u - uεh ‖L_∞(Ω) ≤ C(A,u) h2α/(2 + α) | ln h | 0lt; α≤ 2, provided ε≈ h2/(2+α). Such a convergence rate is at best of order h | ln h |, which turns out to be quasi-optimal.