An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems

TL;DR

This paper introduces a new embedded discontinuous Galerkin (EDG) method for second-order elliptic problems on polygonal/polyhedral meshes, achieving optimal convergence rates through specific polynomial degree choices.

Contribution

The paper develops a novel EDG method with tailored polynomial degrees for potential, trace, and flux, and proves its optimal convergence on complex meshes.

Findings

Achieves optimal convergence rates for potential and flux.

Validates theoretical results with numerical experiments.

Applicable to polygonal/polyhedral meshes.

Abstract

The embedded discontinuous Galerkin (EDG) method by Cockburn et al. [SIAM J. Numer. Anal., 2009, 47(4), 2686-2707] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees $k+1$, $k+1$, $k$ ($k\geq 0$) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.