A coupling of weak Galerkin and mixed finite element methods for poroelasticity

Abstract

In this paper, we present a coupling of a weak Galerkin method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase in poroelasticity equation. Because our method provides a stable element combination for the mixed linear elasticity problem, it can avoid the poroelasticity locking mathematically. Indeed, uniform-in-time error estimates of all the unknowns are obtained for both semidiscrete scheme and fully discrete scheme without assuming that the constrained specific storage coefficient is uniformly positive. Compared with the method proposed by Jeonghun J. Lee, our method can get the arbitrary convergence order by changing degree of the polynomial. As for the computational cost, we can use the modified weak Galerkin method to eliminate the boundary term and reduce the number of unknowns. In addition, we do not use Gronwall inequality in the error analysis. Finally, the numerical experiments verify the theoretical analysis and show the effectiveness to overcome spurious pressure oscillations.