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

Weifeng Qiu

TL;DR

The article presents a unified finite element framework for both linear elliptic PDEs in non-divergence form and the elliptic HJB equation on Lipschitz polyhedra in 2D and 3D. It establishes global $W^{2,p}$ regularity and discrete $W_h^{2,p}$ convergence under either continuous coefficients or Cordes-type conditions for discontinuous coefficients, and extends the theory to HJB with potentially uncountable index sets via measurability and a contraction argument. A key feature is the gamma-weighted reformulation and the discrete gradient operator, enabling stability and well-posedness analyses across a broad range of $p$ and domain geometries, including non-convex 2D polygons. The results provide high-regularity, optimal convergence guarantees beyond existing low-order or divergence-form approaches and unify the treatment of linear non-divergence PDEs and HJB equations within a single FEM framework.

Abstract

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the elliptic Hamilton-Jacobi-Bellman (HJB) equation. For the linear elliptic PDE in non-divergence form, we consider two scenarios of the matrix coefficient matrix $A$. One is $A$ is uniformly continuous. The other is $A$ is discontinuous but $γA$ is dominated by $I_{d}$ where $γ$ is a positive weight function. We prove that optimal convergence in discrete $W^{2,p}$-norm of the numerical 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 neighbourhood of $\frac{4}{3}$. We also prove the well-posedness of strong solution in $W^{2,p}(Ω)$ for both linear elliptic PDE in non-divergence form and the HJB equation for $1< p \leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$) and for $p$ in an open interval starting from $1$ and including $\frac{4}{3}$ on two dimensional non-convex polygon. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

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

TL;DR

The article presents a unified finite element framework for both linear elliptic PDEs in non-divergence form and the elliptic HJB equation on Lipschitz polyhedra in 2D and 3D. It establishes global regularity and discrete convergence under either continuous coefficients or Cordes-type conditions for discontinuous coefficients, and extends the theory to HJB with potentially uncountable index sets via measurability and a contraction argument. A key feature is the gamma-weighted reformulation and the discrete gradient operator, enabling stability and well-posedness analyses across a broad range of and domain geometries, including non-convex 2D polygons. The results provide high-regularity, optimal convergence guarantees beyond existing low-order or divergence-form approaches and unify the treatment of linear non-divergence PDEs and HJB equations within a single FEM framework.

Abstract

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the elliptic Hamilton-Jacobi-Bellman (HJB) equation. For the linear elliptic PDE in non-divergence form, we consider two scenarios of the matrix coefficient matrix . One is is uniformly continuous. The other is is discontinuous but is dominated by where is a positive weight function. We prove that optimal convergence in discrete -norm of the numerical approximation to the strong solution for on convex polyhedra in (). If the domain is a two dimensional non-convex polygon, is valid in a neighbourhood of . We also prove the well-posedness of strong solution in for both linear elliptic PDE in non-divergence form and the HJB equation for on convex polyhedra in () and for in an open interval starting from and including on two dimensional non-convex polygon. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

Paper Structure

This paper contains 22 sections, 21 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Main contributions
  4. Contributions for (\ref{['nondiv_pde_original']})
  5. Contributions for (\ref{['hjb_eqs_original']})
  6. Structure of this paper
  7. Assumptions, notations, FEM and preliminary results
  8. Mesh and space notation
  9. Finite element method for linear elliptic PDE in non-divergence form and HJB equation
  10. FEM for linear elliptic PDE in non-divergence form
  11. FEM for the Hamilton-Jacobi-Bellman equation
  12. Preliminary results
  13. Linear elliptic PDE in non-divergence form
  14. Global $W^{2,p}$ estimate of linear elliptic PDE in non-divergence form with uniformly continuous $A$
  15. Stability analysis of FEM for linear elliptic PDE in non-divergence form with uniformly continuous $A$
  16. ...and 7 more sections

Key Result

Lemma 2.1

If the domain $\Omega$ is a convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$), let $p \in (1,2]$. If the domain is a Lipschitz polygon in $\mathbb{R}^{2}$, let $p \in (1,\frac{4}{3} + \epsilon_{2})$ where $\epsilon_{2}>0$ depends only on $\Omega$. Then for any $g \in L^{p}(\Omega)$, there is a unique s Here the constant $\underline{C}_{p}$ may depend on $p$. If the domain $\Omega$ is a convex polyhed

Theorems & Definitions (43)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Remark 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 33 more