A finite element method for second order nonvariational elliptic problems

TL;DR

This paper introduces a finite element method tailored for second order elliptic problems in nonvariational form, utilizing a finite element Hessian and Schur complement techniques, demonstrated through computational experiments.

Contribution

It presents a novel finite element approach specifically designed for nonvariational elliptic problems, including the development of a finite element Hessian concept.

Findings

Effective approximation of nonvariational elliptic problems demonstrated

Method successfully applied to linear and quasilinear PDEs

Computational experiments validate the approach

Abstract

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of a 'finite element Hessian' and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasilinear PDE, all in nonvariational form.