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

Abstract

Summary We consider an efficient preconditioner for a boundary integral equation (BIE) formulation 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 generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of H2‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as scriptOfalse(N2ptnormallnormalonormalg21false/εfalse). 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.

A boundary-integral framework to simulate viscous erosion of a porous medium

“…In this work, we consider geometries requiring O(100) GMRES iterations, so preconditioners are currently unnecessary. In future work, we will address mobile bodies for which near contacts may necessitate preconditioners [19,58,60]. Similarly, near-singular schemes will not be necessary due to the fact that the spacing between receding bodies of fixed position always grows in time.…”

Parallelization of the inverse fast multipole method with an application to boundary element method

“…This improved the timing result in practice. 13 2 shows the statistics of the hierarchies used in the following computations. We can observe that the number of nodes of the FMM and IFMM (in the second and fourth columns, respectively) is ∼ 4 κ at level κ, which is because the distribution of boundary elements is planar in 3D.…”

“…This allows the IFMM to solve 1 with the same asymptotic computation and memory complexities as the FMM, i.e., O(N log 2 1 ε ), where ε is a prescribed accuracy. The efficiency of the IFMM has been studied in several previous works for solving 1: Quaife et al [13] and Coulier et al [6] applied the IFMM as preconditioners for the GMRES [14] to solve 1 from the immersed boundary method regarding Stokes flow problems in two and three dimensions (3D); Takahashi et al [15] applied the IFMM together with the low-frequency FMM to accelerate the BEM for the 3D Helmholtz equation; Coulier et al [16] applied the IFMM to reduce the cost of a mesh deformation method which is based on the radial basis function interpolation.…”

Stable and contact‐free time stepping for dense rigid particle suspensions

“…Therefore, preconditioners are often applied. There are a variety of preconditioners available for integral equations [13,16,18,31,64,66], we apply a simple block-diagonal preconditioner that was successfully used for vesicle suspensions [62].…”