Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-Divergence Form Elliptic PDEs

TL;DR

This paper introduces interior penalty discontinuous Galerkin methods for second order linear elliptic PDEs in non-divergence form, proving their optimal convergence and providing numerical validation.

Contribution

The paper develops a novel IP-DG method for non-divergence form elliptic PDEs, establishing convergence and error estimates with a new discrete Calderon-Zygmund estimate.

Findings

Optimal convergence in a discrete $W^{2,p}$-norm.

Establishment of a discrete Calderon-Zygmund estimate.

Numerical experiments confirming theoretical results.

Abstract

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $W^{2,p}$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the $W^{2,p}$ strong solution in a discrete $W^{2,p}$-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon-Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. As a byproduct of our analysis, we also establish broken $W^{1,p}$-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.