A comparison among Vanka, Uzawa and Fixed-Stress smoothers for the one-dimensional poroelasticity problem using Multigrid Time-Stepping

Resumo

The poroelasticity equations mathematically model the interaction between the deformation of a porous
elastic material and the fluid flow inside it. The mathematical model that describes this theory, in its most simplified
version, considers the variables displacement, pressure and time, related to each other by a system of partial
differential equations. The importance of deepening the knowledge about this problem is related to the difficulty
of obtaining a numerical solution, due to the presence of saddle points that generates instability in the numerical
analysis. In this work, the problem of 1D poroelasticity is solved, whose boundary conditions assume a left
boundary without displacement variation and with free drainage, and a rigid right border without pressure
variation. For the discretization of differential equations, the Finite Volume Method is used for spatial
discretization and the implicit Euler method with Time-Stepping sweep for temporal discretization are used. The
linear systems from discretization are solved using Vanka, Uzawa and Fixed-Stress smoothers. The results
obtained demonstrate that the different smoothers are equivalent with respect to accuracy. However, there are
differences regarding the convergence factor, computational time and complexity.