Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems

Radley, Thomas J.; Houston, Paul; Hubbard, Matthew E.

Abstract

In this article we consider the application of Euler’s homogeneous function theorem to- gether with Stokes’ theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two and three dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its deriva- tives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to com- pute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.

Citation

Radley, T. J., Houston, P., & Hubbard, M. E. (2024). Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems. Mathematics in Engineering, 6(1), 192-220

• Journal Article Type: Article
• Acceptance Date: Feb 14, 2024
• Online Publication Date: Mar 5, 2024
• Publication Date: Mar 5, 2024
• Deposit Date: Feb 21, 2024
• Publicly Available Date: Mar 8, 2024
• Journal: Mathematics in Engineering
• Electronic ISSN: 2640-3501
• Publisher: AIMS Press
• Peer Reviewed: Peer Reviewed
• Volume: 6
• Issue: 1
• Pages: 192-220
• Keywords: Polytopic elements; numerical integration; discontinuous Galerkin methods
• Public URL: https://nottingham-repository.worktribe.com/output/31610287