Orthogonal bases for vertex-mapped pyramids

TL;DR

This paper introduces an efficient explicit Discontinuous Galerkin method for pyramidal meshes using a semi-nodal orthogonal basis, improving computational efficiency despite non-affine mappings.

Contribution

It develops a semi-nodal orthogonal basis for pyramids that simplifies mass matrix inversion in DG methods on non-affine meshes.

Findings

Confirmed expected convergence rates through numerical experiments.

Demonstrated improved efficiency in DG computations with the new basis.

Validated the orthogonality and suitability of the basis for pyramidal elements.

Abstract

Discontinuous Galerkin (DG) methods discretized under the method of lines must handle the inverse of a block diagonal mass matrix at each time step. Efficient implementations of the DG method hinge upon inexpensive and low-memory techniques for the inversion of each dense mass matrix block. We propose an efficient time-explicit DG method on meshes of pyramidal elements based on the construction of a semi-nodal high order basis, which is orthogonal for a class of transformations of the reference pyramid, despite the non-affine nature of the mapping. We give numerical results confirming both expected convergence rates and discuss efficiency of DG methods under such a basis.