Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form A(x) : D²u(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 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-uhϵ‖L≤C(A,u)h2α/(2+α)|lnh|02,provided ϵ≈ h^2/(2+α). Such a convergence rate is at best of order h| ln h| , which turns out to be quasi-optimal.