$hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes
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arXiv
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| Natura: | Preprint |
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2016
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| author | Cangiani, Andrea Dong, Zhaonan Georgoulis, Emmanuil H. |
| author_facet | Cangiani, Andrea Dong, Zhaonan Georgoulis, Emmanuil H. |
| contents | We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of \emph{total} degree, say $p$, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete $hp$-dG scheme using less degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with \emph{arbitrary} number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1605_01212 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | $hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes Cangiani, Andrea Dong, Zhaonan Georgoulis, Emmanuil H. Numerical Analysis We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of \emph{total} degree, say $p$, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete $hp$-dG scheme using less degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with \emph{arbitrary} number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments. |
| title | $hp$-Version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/1605.01212 |