Stabilized finite difference methods for the fully dynamic biot's problem

This paper deals with the stabilization of the poroelasticity system, in the incompressible fully dynamic case. The stabilization term is a perturbation of the equilibrium equation that allows us to use central difference schemes to approximate the first order spatial derivatives, yielding numerical solutions without oscillations independently of the chosen discretization parameters. The perturbation term is a discrete Laplacian of the forward time difference, affected by a stabilization parameter depending on the mesh size and the properties of the porous medium. In the one dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented to show the efficiency of the proposed stabilization technique

Finite-difference analysis for the linear thermoporoelasticity problem and its numerical resolution by multigrid methods

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence