Automatic symbolic computation for discontinuous Galerkin finite element methods

Houston, Paul; Sime, Nathan

Abstract

The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Frechet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems.

Citation

Houston, P., & Sime, N. (2018). Automatic symbolic computation for discontinuous Galerkin finite element methods. SIAM Journal on Scientific Computing, 40(3), Article C327-C357. https://doi.org/10.1137/17M1129751

• Journal Article Type: Article
• Acceptance Date: Mar 13, 2018
• Online Publication Date: Jun 12, 2018
• Publication Date: Jul 1, 2018
• Deposit Date: Mar 15, 2018
• Publicly Available Date: Jun 12, 2018
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1095-7197
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 40
• Issue: 3
• Article Number: C327-C357
• DOI: https://doi.org/10.1137/17M1129751
• Keywords: Symbolic computation, finite element methods, discontinuous Galerkin methods
• Public URL: https://nottingham-repository.worktribe.com/output/943433
• Publisher URL: https://epubs.siam.org/doi/10.1137/17M1129751
• Contract Date: Mar 15, 2018