Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods

Understand

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.