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An H(div)-conforming finite element method for the Biot’s consolidation mo- del is developed, with displacements and fluid velocity approximated by elements from BDM_k space. The use of H(div)-conforming elements for flow variables ensures the local mass conservation. In the H(div)-conforming approximation of displacement, the tan- gential components are discretised in the interior penalty discontinuous Galerkin frame- work,and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully dis- crete scheme,prove their unique solvability and establish optimal error estimates for each variable.
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A cutFEM divergence–free discretization for the stokes problem
We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O ( h ) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O ( h −1 ), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.
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- PAR ID:
- 10418213
- Date Published:
- Journal Name:
- ESAIM: Mathematical Modelling and Numerical Analysis
- Volume:
- 57
- Issue:
- 1
- ISSN:
- 2822-7840
- Page Range / eLocation ID:
- 143 to 165
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1051/m2an/2022072
