Francisco J. Gaspar

TL;DR: It is shown that even in 1D a Stokes-stable finite element pair fails to provide a monotone discretization for the pressure in such regimes, and a stabilization term is introduced which removes the oscillations.

TL;DR: In this article, stability estimates and convergence analysis of finite difference methods for the Biot's consolidation model are presented, and central differences for space discretization and a weighed two-level time scheme are analyzed.

TL;DR: In this article, a stable finite element scheme that avoids pressure oscillations for a three-field Biot's model in poroelasticity is considered, and the involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix-Raviart finite elements for the displacement, lowest order Raviart-Thomas-Nedelecźelements for the Darcy velocity, and piecewise constant approximation for the pressure.

TL;DR: In this article, a stabilized finite element scheme for the poroelasticity equations is proposed, based on the perturbation of the flow equation, allowing us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters.

TL;DR: New multigrid line smoothers for the solution of higher order discretizations of convection-dominated problems directly are presented and evaluated theoretically with Fourier smoothing and two-grid analysis.