Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

TL;DR

This paper introduces a finite element method for second order elliptic PDEs in non-divergence form and HJB equations, proving well-posedness and optimal convergence under relaxed coefficient continuity assumptions.

Contribution

It develops a unified finite element approach applicable to both PDE types, with new convergence proofs and relaxed coefficient regularity requirements.

Findings

Proves well-posedness of strong solutions in $W^{2,p}$ spaces.

Establishes optimal convergence rates in discrete $W^{2,p}$-norm.

Extends applicability to non-convex polygons and less regular coefficients.

Abstract

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(\Omega)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.