Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem
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- by Christoph Lehrenfeld, Tim van Beeck and Igor Voulis;
- Math. Comp. 94 (2025), 1667-1699
- DOI: https://doi.org/10.1090/mcom/4027
- Published electronically: October 30, 2024
- HTML | PDF
Abstract:
In this paper we present a new $H(\operatorname {div})$-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.References
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Bibliographic Information
- Christoph Lehrenfeld
- Affiliation: Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
- MR Author ID: 984763
- ORCID: 0000-0003-0170-8468
- Email: lehrenfeld@math.uni-goettingen.de
- Tim van Beeck
- Affiliation: Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
- ORCID: 0009-0002-4550-8636
- Email: t.beeck@math.uni-goettingen.de
- Igor Voulis
- Affiliation: Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
- MR Author ID: 1305472
- Email: i.voulis@math.uni-goettingen.de
- Received by editor(s): June 22, 2023
- Received by editor(s) in revised form: April 5, 2024, and July 10, 2024
- Published electronically: October 30, 2024
- © Copyright 2024 by the authors
- Journal: Math. Comp. 94 (2025), 1667-1699
- MSC (2020): Primary 65N12, 65N30; Secondary 65N85
- DOI: https://doi.org/10.1090/mcom/4027
- MathSciNet review: 4888019