A mixed virtual element method for Biot's consolidation model

In this paper, we shall present a weak virtual element method for the standard three field poroelasticity problem on polytopal meshes. The flux velocity and pressure are approximated by the low order $H\left(\text{div}\right)$ virtual element and the piecewise constant, while the elastic displacement is discretized by the $H\left(\text{div}\right)$ virtual element with some tangential polynomials on element boundaries. A fully discrete scheme is then given by choosing the backward Euler for the time discretization. With some assumptions on the exact solutions, we prove that the convergence order is of order 1 with respect to the meshsize and the time step, and the hidden constants are independent of the parameters of the problems. Some numerical experiments are given to verify the results.