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) : D^2 u(x) = f(x)) in a bounded but not necessarily convex domain (Omega ) and study it in the max norm. The fine scale is given by the meshsize h, whereas the coarse scale (epsilon ) 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

$$begin{aligned} Vert ,u - u^{epsilon }_h,Vert _{L^{infty }(Omega )} le , C(A,u) , h^{2alpha /(2 + alpha )} big | ln h big | qquad 0< alpha le 2, end{aligned}$$

provided (epsilon approx h^{2/(2+alpha )}). Such a convergence rate is at best of order ( h big | ln h big |), which turns out to be quasi-optimal.