DG = FEM + flat elements, Part I: Diffusion
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Szotkowski, Jiří Kučera, Václav Shu, Chi-Wang Quiriny, Antoine Lambrechts, Jonathan Moës, Nicolas Remacle, Jean-François |
| author_facet | Szotkowski, Jiří Kučera, Václav Shu, Chi-Wang Quiriny, Antoine Lambrechts, Jonathan Moës, Nicolas Remacle, Jean-François |
| contents | We establish a simple, rigorous, and easy to implement connection between the classical continuous finite element method (FEM) and the discontinuous Galerkin (DG) method for Poisson's problem. The key idea is to insert a vanishing-thickness layer of "dummy" elements along cell interfaces. By modifying the diffusion coefficient on these elements to be proportional to their thickness, we prove the FEM formulation converges to Babuška-Zlámal DG with trapezoidal edge quadrature. The scheme is trivial to implement by (i) a mesh edit that introduces degenerate interface elements and (ii) a single Jacobian threshold in an otherwise unmodified FEM code to handle the degenerate elements via the tempered finite element (TFEM) framework. We provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal $H^1$ and $L^2$ error estimates, and present numerical experiments in 2D and 3D. The method allows for simple implementation of DG in a FEM code and even adaptive element-by-element switching between FEM and DG with minimal coding effort. The framework is readily extensible, as we will demonstrate in a companion paper dedicated to evolutionary nonlinear first-order hyperbolic systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19037 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | DG = FEM + flat elements, Part I: Diffusion Szotkowski, Jiří Kučera, Václav Shu, Chi-Wang Quiriny, Antoine Lambrechts, Jonathan Moës, Nicolas Remacle, Jean-François Numerical Analysis 65N30, 65N12, 35J05 We establish a simple, rigorous, and easy to implement connection between the classical continuous finite element method (FEM) and the discontinuous Galerkin (DG) method for Poisson's problem. The key idea is to insert a vanishing-thickness layer of "dummy" elements along cell interfaces. By modifying the diffusion coefficient on these elements to be proportional to their thickness, we prove the FEM formulation converges to Babuška-Zlámal DG with trapezoidal edge quadrature. The scheme is trivial to implement by (i) a mesh edit that introduces degenerate interface elements and (ii) a single Jacobian threshold in an otherwise unmodified FEM code to handle the degenerate elements via the tempered finite element (TFEM) framework. We provide a rigorous derivation of the resulting TFEM-DG scheme, prove optimal $H^1$ and $L^2$ error estimates, and present numerical experiments in 2D and 3D. The method allows for simple implementation of DG in a FEM code and even adaptive element-by-element switching between FEM and DG with minimal coding effort. The framework is readily extensible, as we will demonstrate in a companion paper dedicated to evolutionary nonlinear first-order hyperbolic systems. |
| title | DG = FEM + flat elements, Part I: Diffusion |
| topic | Numerical Analysis 65N30, 65N12, 35J05 |
| url | https://arxiv.org/abs/2605.19037 |