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.
Highlights
Many problems in physics involve the numerical solution of second-order elliptic partial differential equations (PDEs)
Many properties that make Discontinuous Galerkin (DG) methods advantageous for time evolutions apply to elliptic problems, which lead to the development of DG schemes for elliptic PDEs [5,6]
DG schemes provide a flexible mechanism for refining the computational grid, retaining exponential convergence even in the presence of discontinuities when adaptive mesh-refinement (AMR) techniques are employed [7,8]
Summary
Many problems in physics involve the numerical solution of second-order elliptic partial differential equations (PDEs). Such elliptic problems often represent static field configurations under the effect of external forces and arise, for example, in electrodynamics, in linear or nonlinear elasticity, and in general relativity. The elliptic DG scheme presented in this article is not limited to applications in numerical relativity It applies to all second-order elliptic problems that can be formulated in first-order flux form. The test problems include scenarios derived from general relativity that feature sets of coupled, strongly nonlinear equations on a curved manifold with nonlinear boundary conditions, solved on curved meshes.
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