Performance Analysis of Block AMG Preconditioning of Poroelasticity Equations

Abstract

The goal of this study is to develop, analyze, and implement efficient numerical algorithms for equations of linear poroelasticity, a macroscopically diphasic description of coupled flow and mechanics. We suppose that the solid phase is governed by the linearized constitutive relationship of Hooke’s law. Assuming in addition a quasi-steady regime of the fluid structure interaction, the media is described by the Biot’s system of equations for the unknown displacements and pressure ($\mathbf{u},p$). A mixed Finite Element Method (FEM) is applied for discretization. Linear conforming elements are used for the displacements. Following the approach of Arnold-Brezzi, non-conforming FEM approximation is applied for the pressure where bubble terms are added to guarantee a local mass conservation. Block-diagonal preconditioners are used for iterative solution of the arising saddle-point linear algebraic system. The BiCGStab and GMRES are the basic iterative schemes, while algebraic multigrid (AMG) is utilized for approximation of the diagonal blocks. The HYPRE implementations of BiCGStab, GMRES and AMG (BoomerAMG, [6]) are used in the presented numerical tests. The aim of the performance analysis is to improve both: (i) the convergence rate of the solvers measured by the iteration counts, and (ii) the CPU time to solve the problem. The reported results demonstrate some advantages of GMRES for the considered real-life, large-scale, and strongly heterogeneous test problems. Significant improvement is observed due to tuning of the BoomerAMG settings.