An efficient preconditioner for the fast simulation of a 2D Stokes flow in porous media

TL;DR

This paper introduces an efficient preconditioner using the inverse fast multipole method to accelerate the solution of boundary integral equations for 2D Stokes flow in porous media, reducing computational cost and iteration count.

Contribution

The paper presents a novel application of the inverse fast multipole method as a preconditioner for boundary integral equations in porous media flow simulations, improving efficiency.

Findings

Preconditioner significantly accelerates GMRES convergence.

Method maintains high accuracy with tunable parameter ε.

Demonstrated effectiveness on pipe flow problems with many pores.

Abstract

We consider an efficient preconditioner for boundary integral equation (BIE) formulations of the two-dimensional Stokes equations in porous media. While BIEs are well-suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as GMRES. In this paper, we apply a fast inexact direct solver, the inverse fast multipole method (IFMM), as an efficient preconditioner for GMRES. This solver is based on the framework of $\mathcal{H}^{2}$-matrices and uses low-rank compressions to approximate certain matrix blocks. It has a tunable accuracy $\varepsilon$ and a computational cost that scales as $\mathcal{O} (N \log^2 1/\varepsilon)$. We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block-diagonal preconditioner, especially for pipe flow problems involving many pores.