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A multipoint stress-flux mixed finite element method for the Stokes-Biot model

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Abstract

In this paper we present and analyze a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of stresses, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, where the symmetry of the Stokes and poroelastic stress tensors is imposed by setting the vorticity and structure rotation tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly by introducing the traces of the fluid velocity, structure velocity, and the poroelastic media pressure on the interface as the associated Lagrange multipliers. The existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with non-matching grids, together with the corresponding stability bounds. In addition, we develop a new multipoint stress-flux mixed finite element method by involving the vertex quadrature rule, which allows for local elimination of the stresses, rotations, and Darcy fluxes. Well-posedness and error analysis with corresponding rates of convergence for the fully-discrete scheme are complemented by several numerical experiments.

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Authors and Affiliations

  1. Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Sergio Caucao

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Tongtong Li & Ivan Yotov

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  1. Sergio Caucao
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  2. Tongtong Li
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  3. Ivan Yotov
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Correspondence to Ivan Yotov.

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Sergio Caucao: Supported in part by ANID-Chile through the project PAI77190084 of the PAI Program: Convocatoria Nacional Subvención a la Instalación en la Academia (convocatoria 2019) and Department of Mathematics, University of Pittsburgh.

Tongtong Li and Ivan Yotov: Supported in part by NSF grants DMS 1818775 and DMS 2111129.

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Caucao, S., Li, T. & Yotov, I. A multipoint stress-flux mixed finite element method for the Stokes-Biot model. Numer. Math. 152, 411–473 (2022). https://doi.org/10.1007/s00211-022-01310-2

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