Fully-Mixed Virtual Element Method for the Biot Problem
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916706361802752 |
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| author | Botti, Michele Prada, Daniele Scotti, Anna Visinoni, Michele |
| author_facet | Botti, Michele Prada, Daniele Scotti, Anna Visinoni, Michele |
| contents | Poroelasticity describes the interaction of deformation and fluid flow in saturated porous media. A fully-mixed formulation of Biot's poroelasticity problem has the advantage of producing a better approximation of the Darcy velocity and stress field, as well as satisfying local mass and momentum conservation. In this work, we focus on a novel four-fields Virtual Element discretization of Biot's equations. The stress symmetry is strongly imposed in the definition of the discrete space, thus avoiding the use of an additional Lagrange multiplier. A complete a priori analysis is performed, showing the robustness of the proposed numerical method with respect to limiting material properties. The first order convergence of the lowest-order fully-discrete numerical method, which is obtained by coupling the spatial approximation with the backward Euler time-advancing scheme, is confirmed by a complete 3D numerical validation. A well known poroelasticity benchmark is also considered to assess the robustness properties and computational performance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fully-Mixed Virtual Element Method for the Biot Problem Botti, Michele Prada, Daniele Scotti, Anna Visinoni, Michele Numerical Analysis 65M12, 65M60, 74F10, 76S05 Poroelasticity describes the interaction of deformation and fluid flow in saturated porous media. A fully-mixed formulation of Biot's poroelasticity problem has the advantage of producing a better approximation of the Darcy velocity and stress field, as well as satisfying local mass and momentum conservation. In this work, we focus on a novel four-fields Virtual Element discretization of Biot's equations. The stress symmetry is strongly imposed in the definition of the discrete space, thus avoiding the use of an additional Lagrange multiplier. A complete a priori analysis is performed, showing the robustness of the proposed numerical method with respect to limiting material properties. The first order convergence of the lowest-order fully-discrete numerical method, which is obtained by coupling the spatial approximation with the backward Euler time-advancing scheme, is confirmed by a complete 3D numerical validation. A well known poroelasticity benchmark is also considered to assess the robustness properties and computational performance. |
| title | Fully-Mixed Virtual Element Method for the Biot Problem |
| topic | Numerical Analysis 65M12, 65M60, 74F10, 76S05 |
| url | https://arxiv.org/abs/2504.17729 |