Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form

TL;DR

This paper introduces a finite element method for approximating solutions to second-order elliptic PDEs in nondivergence form, utilizing the ABP maximum principle for error control and demonstrating superior adaptive performance.

Contribution

It develops a novel FEM approach that leverages the ABP maximum principle for error estimation, extending to nonconforming elements and enabling adaptive refinement.

Findings

Convergence established for uniform mesh-refinements.

The method provides reliable a posteriori error estimates.

Adaptive computations outperform uniform refinement in benchmarks.

Abstract

This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm.