$hp$-Version robust interior penalty discontinuous Galerkin methods for the $p$-Laplacian on simplicial and on essentially arbitrarily-shaped element meshes
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908973807960064 |
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| author | Georgoulis, Emmanuil H. Paraschis, Panagiotis |
| author_facet | Georgoulis, Emmanuil H. Paraschis, Panagiotis |
| contents | We consider the discretization of the $p$-Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to unconditional stability of the method. Further, $hp$-version a priori norm and quasi-norm error estimates are established, subordinate to available polynomial approximation results. The analysis is extended to discontinuous Galerkin methods, based on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements. This extension requires the proof of new $hp$-version weighted inverse estimates on essentially arbitrarily-shaped elements. Numerical experiments are also presented, highlighting the relevance of the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $hp$-Version robust interior penalty discontinuous Galerkin methods for the $p$-Laplacian on simplicial and on essentially arbitrarily-shaped element meshes Georgoulis, Emmanuil H. Paraschis, Panagiotis Numerical Analysis 65N30, 35J92, 65J15 We consider the discretization of the $p$-Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to unconditional stability of the method. Further, $hp$-version a priori norm and quasi-norm error estimates are established, subordinate to available polynomial approximation results. The analysis is extended to discontinuous Galerkin methods, based on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements. This extension requires the proof of new $hp$-version weighted inverse estimates on essentially arbitrarily-shaped elements. Numerical experiments are also presented, highlighting the relevance of the theoretical findings. |
| title | $hp$-Version robust interior penalty discontinuous Galerkin methods for the $p$-Laplacian on simplicial and on essentially arbitrarily-shaped element meshes |
| topic | Numerical Analysis 65N30, 35J92, 65J15 |
| url | https://arxiv.org/abs/2604.15879 |