On an Uzawa smoother in multigrid for poroelasticity equations

Abstract

SummaryA poroelastic saturated medium can be modeled by means of Biot's theory of consolidation. It describes the time-dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure. It shall be interpreted as a combination of the symmetric Gauss-Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme.

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

“…So, this variant is the one that we extend to the Darcy equation. The efficiency of the proposed Uzawa smoother for threedimensional Stokes and Biot poroelasticity equations was presented in [23] and [34], respectively. In addition, M A has two important properties needed in our theoretical analysis, in addition to its efficiency as a smoother.…”

“…As is commonly done, in this work, we mainly focus on the case of constant hydraulic conductivity K. The proposed multigrid solution method can, however, be generalized to varying K values and also to the case where the hydraulic conductivity is prescribed by a full tensor K. In [34], we applied a variant of the Uzawa smoother for porous media flow when anisotropies due to grid stretching appeared. Also, heterogeneous coefficients were considered in that work.…”

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

“…The Uzawa smoother was proposed for Stokes problems in fluid dynamics in [11], [17]. In [12], a multigrid method with an Uzawa smoother was successfully applied for solving poroelastic equations. Thus, it seems natural to assume that this relaxation is also suitable for the coupled Stokes-poroelasticity problem.…”

“…Optimal relaxation parameter In [12] and [17], different analytic expressions of ω for the individual Stokes and poroelastic systems are obtained by a theoretical analysis that we can use directly for the coupled system as well. Local Fourier analysis is a powerful tool for the quantitative analysis of multigrid.…”

“…It helps to predict the asymptotic convergence factor of the multigrid algorithm and provides a concrete formula for the optimal relaxation parameter. In [12], the optimal parameter for the poroelastic system is given by…”