Time-continuous strongly conservative space-time finite element methods for the dynamic Biot model
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| Format: | Preprint |
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2025
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| author | Kraus, Johannes Lymbery, Maria Osthues, Kevin |
| author_facet | Kraus, Johannes Lymbery, Maria Osthues, Kevin |
| contents | We consider the dynamic Biot model (see [Biot, M. A. J. Appl. Phys. 33, 1482--1498 (1962)]) describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. This model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest.
A family of variational space-time finite element methods is proposed, which combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with $H(\mathrm{div})$-conforming approximations of the displacement field, its time derivative, and the flux field--of discontinuous Galerkin (DG) type for displacements--with a piecewise polynomial pressure approximation, providing an inf-sup stable strongly conservative mixed method in each case. We prove error estimates in a combined energy norm in space for the maximum norm in time. The theoretical results are confirmed by numerical experiments for different polynomial orders in space and time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_19955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time-continuous strongly conservative space-time finite element methods for the dynamic Biot model Kraus, Johannes Lymbery, Maria Osthues, Kevin Numerical Analysis 74H15 65M15 65M60 74F10 We consider the dynamic Biot model (see [Biot, M. A. J. Appl. Phys. 33, 1482--1498 (1962)]) describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. This model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest. A family of variational space-time finite element methods is proposed, which combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with $H(\mathrm{div})$-conforming approximations of the displacement field, its time derivative, and the flux field--of discontinuous Galerkin (DG) type for displacements--with a piecewise polynomial pressure approximation, providing an inf-sup stable strongly conservative mixed method in each case. We prove error estimates in a combined energy norm in space for the maximum norm in time. The theoretical results are confirmed by numerical experiments for different polynomial orders in space and time. |
| title | Time-continuous strongly conservative space-time finite element methods for the dynamic Biot model |
| topic | Numerical Analysis 74H15 65M15 65M60 74F10 |
| url | https://arxiv.org/abs/2507.19955 |