A five-field mixed formulation for the fully dynamic Biot-Brinkman model

We propose and analyze a mixed formulation for the fully dynamic Biot--Brinkman system, which models the coupled effects of internal viscous diffusion and deformation of the porous skeleton. The classical formulation is expressed in terms of displacement, fluid velocity, and pore pressure. In this work, we reformulate the problem in terms of the fluid velocity, pore pressure, and poroelastic stress tensor, together with two additional variables: the structural velocity and the rotation rate. In particular, we eliminate the solid displacement and the rotation tensor from the set of primary variables. Both quantities are subsequently recovered by a simple post-processing step. This yields a five-field mixed formulation. We establish existence and uniqueness of a weak solution and derive stability bounds based on the Babu\v{s}ka--Brezzi theory for perturbed saddle-point problems, as well as on the theoretical framework developed by Showalter for semilinear degenerate evolution equations. We then develop a semidiscrete continuous-in-time approximation based on stable mixed finite elements. Moreover, employing a backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove that both schemes are well-posed, derive stability bounds, and establish the corresponding error estimates. Finally, we numerically verify the theoretical convergence rates and perform classical benchmark tests to illustrate the robustness and versatility of the method across different domains and parameter regimes.