Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917766732185600 |
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| author | Lehrenfeld, Christoph van Beeck, Tim Voulis, Igor |
| author_facet | Lehrenfeld, Christoph van Beeck, Tim Voulis, Igor |
| contents | In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem Lehrenfeld, Christoph van Beeck, Tim Voulis, Igor Numerical Analysis In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results. |
| title | Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2306.12722 |