Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs
In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear pr...
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Springer Netherlands
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| Online Access: | https://eprints.nottingham.ac.uk/1493/ |
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| author | Congreve, Scott Houston, Paul Wihler, Thomas P. |
| author_facet | Congreve, Scott Houston, Paul Wihler, Thomas P. |
| author_sort | Congreve, Scott |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V_{H,P}. The resulting `coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V_{h,p}; thereby, only a linear system of equations is solved on the richer space V_{h,p}. In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V_{H,P} and V_{h,p}, respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the cpu time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method. |
| first_indexed | 2025-11-14T18:15:27Z |
| format | Article |
| id | nottingham-1493 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:15:27Z |
| publisher | Springer Netherlands |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-14932020-05-04T20:34:28Z https://eprints.nottingham.ac.uk/1493/ Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs Congreve, Scott Houston, Paul Wihler, Thomas P. In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V_{H,P}. The resulting `coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V_{h,p}; thereby, only a linear system of equations is solved on the richer space V_{h,p}. In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V_{H,P} and V_{h,p}, respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the cpu time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method. Springer Netherlands Article NonPeerReviewed Congreve, Scott, Houston, Paul and Wihler, Thomas P. Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs. Journal of Scientific Computing . ISSN 0885-7474 (Submitted) http://www.springer.com/mathematics/computational+science+%26+engineering/journal/10915 |
| spellingShingle | Congreve, Scott Houston, Paul Wihler, Thomas P. Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title | Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title_full | Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title_fullStr | Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title_full_unstemmed | Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title_short | Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs |
| title_sort | two-grid hp-version discontinuous galerkin finite element methods for second-order quasilinear elliptic pdes |
| url | https://eprints.nottingham.ac.uk/1493/ https://eprints.nottingham.ac.uk/1493/ |