Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods

TL;DR

This paper introduces two novel methods for applying Galerkin difference basis functions to complex geometries in discontinuous Galerkin methods, ensuring stability and accuracy through innovative coupling and metric treatment.

Contribution

It proposes non-conforming curvilinear GD elements and coupling with simplicial elements, with stability proofs and practical conditions for complex geometries.

Findings

Schemes are energy stable even with variational crimes.

Numerical experiments confirm stability and accuracy.

Coupled schemes effectively handle complex geometries.

Abstract

The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using the tensor product constructions to quadrilateral elements for discretizing partial differential equations. Here we propose two approaches to handling complex geometries using the GD basis within a discontinuous Galerkin finite element setting: (1) using non-conforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial elements. In both cases the (semidiscrete) discontinuous Galerkin method is provably energy stable even when variational crimes are committed and in both cases a weight-adjusted mass matrix is used, which ensures that only the reference mass matrix must be inverted. Additionally, we give sufficient conditions on the treatment of metric terms for the curvilinear, nonconforming GD elements to ensure that the scheme is both constant preserving and conservative. Numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes.