Automatic symbolic computation for discontinuous Galerkin finite element methods
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. T...
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Society for Industrial and Applied Mathematics
2018
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| Online Access: | https://eprints.nottingham.ac.uk/50449/ |
| _version_ | 1848798254757576704 |
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| author | Houston, Paul Sime, Nathan |
| author_facet | Houston, Paul Sime, Nathan |
| author_sort | Houston, Paul |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:16:51Z |
| format | Article |
| id | nottingham-50449 |
| institution | University of Nottingham Malaysia Campus |
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| last_indexed | 2025-11-14T20:16:51Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-504492020-05-04T19:43:29Z https://eprints.nottingham.ac.uk/50449/ Automatic symbolic computation for discontinuous Galerkin finite element methods Houston, Paul Sime, Nathan 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. Society for Industrial and Applied Mathematics 2018-07-01 Article PeerReviewed Houston, Paul and Sime, Nathan (2018) Automatic symbolic computation for discontinuous Galerkin finite element methods. SIAM Journal on Scientific Computing, 40 (3). C327-C357. ISSN 1064-8275 Symbolic computation finite element methods discontinuous Galerkin methods https://epubs.siam.org/doi/10.1137/17M1129751 doi:10.1137/17M1129751 doi:10.1137/17M1129751 |
| spellingShingle | Symbolic computation finite element methods discontinuous Galerkin methods Houston, Paul Sime, Nathan Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title | Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title_full | Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title_fullStr | Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title_full_unstemmed | Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title_short | Automatic symbolic computation for discontinuous Galerkin finite element methods |
| title_sort | automatic symbolic computation for discontinuous galerkin finite element methods |
| topic | Symbolic computation finite element methods discontinuous Galerkin methods |
| url | https://eprints.nottingham.ac.uk/50449/ https://eprints.nottingham.ac.uk/50449/ https://eprints.nottingham.ac.uk/50449/ |