Stability and convergence analysis of mixed finite element approximations for a Biot-Brinkman model

Many multiphysics processes of fluid-solid interaction within a porous medium can be described by the Biot-Brinkman model to account for the effects of viscosity in fluid flow. By introducing the auxiliary variables, we can transform the original problem into two generalized Stokes equations. The generalized Stokes equations incorporate a built-in mechanism to circumvent the Poisson locking for the continuous Galerkin method. Subsequently, we establish an energy law and provide a priori estimates for the reformulated problem. Well-posedness is demonstrated using the standard Galerkin method in conjunction with a compactness argument. After that, we develop stable mixed finite element algorithms for the reformulated problem. Influenced by Lamé constant λ, we design three finite element pairs for the proposed algorithms and present the corresponding error estimates. Numerical tests are conducted to validate the theoretical results.