Unified discontinuous Galerkin scheme for a large class of elliptic equations

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.

22 pages, 13 figures, 2 tables, published version. Results are reproducible with the ancillary input files

References in corpus (6)

• Binary-black-hole initial data with nearly-extremal spins
• Short note on the mass matrix for Gauss-Lobatto grid points
• A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code
• The dependence of test-mass thermal noises on beam shape in gravitational-wave interferometers
• Initial data for black hole-neutron star binaries, with rotating stars
• Numerically modeling Brownian thermal noise in amorphous and crystalline thin coatings

Cited by in corpus (6)

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• A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code
• Simulating binary black hole mergers using discontinuous Galerkin methods
• Quasistationary hair for binary black hole initial data in scalar Gauss-Bonnet gravity
• High-accuracy numerical models of Brownian thermal noise in thin mirror coatings
• Discontinuous Galerkin scheme for elliptic equations on extremely stretched grids