Unified discontinuous Galerkin scheme for a large class of elliptic equations

TL;DR

This paper introduces a versatile discontinuous Galerkin scheme applicable to a broad class of elliptic equations, capable of handling complex geometries, boundary conditions, and nonlinearities, with demonstrated accuracy and implementation in an open-source code.

Contribution

A unified, flexible DG scheme for diverse elliptic PDEs that simplifies implementation across different problems without equation-specific modifications.

Findings

Accurate results on various test problems

Applicable to linear and nonlinear elliptic equations

Implemented successfully in open-source SpECTRE code

Abstract

We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and nonlinear elliptic partial differential equations. The unified scheme can accommodate all second-order elliptic equations that can be formulated in first-order flux form, encompassing problems in linear elasticity, general relativity, and hydrodynamics, including problems formulated on a curved manifold. It allows for a wide range of linear and nonlinear boundary conditions, and accommodates curved and nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. We demonstrate the accuracy of the scheme for a suite of numerical test problems. The scheme is implemented in the open-source SpECTRE numerical relativity code.