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Automatic symbolic computation for discontinuous Galerkin finite element methods

Houston, Paul; Sime, Nathan

Authors

PAUL HOUSTON paul.houston@nottingham.ac.uk
Professor of Computational and Applied Maths

Nathan Sime



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), 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 1064-8275
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 http://eprints.nottingham.ac.uk/id/eprint/50449
Publisher URL https://epubs.siam.org/doi/10.1137/17M1129751
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf

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