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

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

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.