Weak Galerkin finite element method with the total pressure variable for Biot's consolidation model

In this work, we develop a weak Galerkin method for the three-field Biot's consolidation model. The key idea is to consider the total pressure variable. We employ the stable pair of weak Galerkin finite elements to discretize the displacement and total pressure, and use totally discontinuous weak functions to approximate pressure in a semi-discrete scheme. Then, we give the fully discrete scheme based on the backward Euler method in time. Furthermore, we prove the well-posedness of the numerical schemes and derive the optimal error estimates for three variables in their nature norms. Our theoretical results are independent of the Lamé constant λ and the storage coefficient $c_0$. Finally, some experiments that employ different polynomial degrees and polygonal meshes are presented to demonstrate the efficiency and stability of the weak Galerkin method.