Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media

Abstract

SUMMARYLarge-scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block-structured matrix system arising from single-phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement-based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners.

A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

“…An inexact version of Eq. 20, along with its nonsymmetric cousin P −1 gGS , was shown in [18] to be very robust preconditioners for Biot's equations on a problem with extreme contrasts in the material parameters. Algorithm 1 shows the necessary steps to implement this preconditioner.…”

“…The present authors likewise found SA to be a nearly optimal preconditioner for the discontinuous Poisson pressure problem (see [16]) and to perform well on the similarly structured Schur complement approximation found in Eq. 21 (see [18]). …”

“…While forming the exact Schur complement is infeasible, an approximation that was shown in [18] to perform well for high-contrasting material parameters iŝ…”

“…In [18], the present authors demonstrated the efficacy of a preconditioner based on the exact block decomposition in the serial case, using the Schur complement of the 2 × 2 coefficient matrix. For the individual blocks, an algebraic multigrid (AMG) preconditioner is used.…”

“…Simpler Schur complement block preconditioners have been employed successfully for Biot's equation [25,27]. However, it remains to investigate the parallel efficiency of the more advanced block preconditioners from [18], particularly targeting large jumps in material parameters. This is exactly the topic of the present paper.…”