Applied Mathematics Seminar
Wujun Zhang
University of Maryland
Discrete ABP estimate and rates of convergence of linear elliptic PDEs in non-divergence form
Abstract: We design a finite element method (FEM) for linear elliptic equations
in non-divergence form, which hinges on an integro-differential
approximation of the PDE. We show the FEM satisfies the discrete
maximum principle (DMP) provided that the mesh is weakly acute. Thanks
to the DMP and consistency property of the FEM, we establish
convergence of the numerical solution to the viscosity solution.
We derive a discrete Alexandroff-Bakelman-Pucci (ABP) estimate for
finite element methods. 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 to optimal
rates of convergence for the finite element method under suitable
regularity assumptions on the solution and coefficient matrix.
Monday November 24, 2014 at 4:00 PM in SEO 636