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). The fine scale is given by the meshsize h whereas the coarse scale is dictated by an integrodifferential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) for any uniformly positive definite matrix A(x) provided that the mesh is weakly acute. Combining the DMP with weak operator consistency of the FEM, we establish convergence of the numerical solution u h to the viscosity solution u as , h→ 0 and ≥ Ch| lnh|. We develop a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a geometric interpretation of Alexandroff estimate and control of the measure of the sub-differential of piecewise linear functions in terms of jumps, and thus of the discrete PDE. The discrete ABP estimate leads, under suitable regularity assumptions on A and u, to the estimate ‖u− u h ‖L∞(Ω) ≤ C ( h ∣∣ lnh∣∣)α/(2+α) 0 < α ≤ 2, provided ≈ (h2| lnh|)1/(2+α). Such a convergence rate is at best of order h ∣∣ lnh∣∣1/2, which turns out to be quasi-optimal.