New Geometric Immersed Interface Multigrid Solvers

Abstract

elliptic interface problems using the maximum principle preserving schemes. In this paper, we improve on this method by giving a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse-grid matrices are M-matrices. We compare this new restrictor to injection and the transpose of interpolation. We show that the number of V-cycles is constant as the mesh size decreases and increases only slightly as the ratio of the discontinuous problem coefficient grows at the interface only when this new restrictor is used. Introduction.In [1], a multigrid method was designed specifically for interface problems that have been discretized using the methods described therein, and in [6] for elliptic interface problems using the maximum principle preserving schemes. In this paper, we improve on this method by giving a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse-grid matrices are M-matrices. We compare this new restrictor to injection and the transpose of interpolation. We show that the number of V-cycles is constant as the mesh size h decreases and increases only slightly as the ratio of the discontinuous problem coefficient β(x, y) grows at the interface only when this new restrictor is used.The paper is organized as follows. In section 2, we review the components of a geometric multigrid (MG) solver and state how our methods fit into this framework. In section 3, we give the particular immersed interface problem that we use as a test example and discuss in section 3.1 how scaling this problem can affect the multigrid performance. In section 4, we develop a new restrictor and a better interpolator. We discuss the motivation for this new component selection in general terms in section 4.1. In section 4.2 we develop a new interpolator for fine-grid points near the internal interfaces. In section 4.3, we examine the choice of three different restrictors and discuss the coarse-grid matrices that result from using them in combination with the interpolator in section 4.2. The properties of these coarse-grid matrices are discussed for the case when the matrix A h is an M-matrix, and proofs that the coarser grid matrices are also M-matrices are given in the appendix. Section 5 gives our numerical results that show the effectiveness of the new restrictor and interpolator. We summarize our findings in section 6.

Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems

“…In [1][2][3], multigrid methods were designed specifically for interface problems discretized by immersed interface methods [13,15]. For interfaces with moderate curvature, the method proposed in [1] produced satisfactory performance. AMG is employed in [7] to solve the resulting linear system.…”

“…AMG is employed in [7] to solve the resulting linear system. It is observed that convergence becomes worse when the problem has high-contrast coefficients and thus many more iterations are needed to achieve reasonable accuracy [1].…”

A kernel-free boundary integral method for elliptic boundary value problems

“…For example, if we rewrite the boundary integral equation (19) as (24) with (25) and rewrite the boundary integral equation (21) as (26) with (27) the application of the GMRES iterative method to solving (24) and (26) simply requires computation of ḡ D , ḡ N and the actions φ, *ψ of the operators , * on φ and ψ, respectively. Once again, the right hand sides of (25) and (27) can be approximated by limit values of approximate solutions to the Dirichlet BVP while direct evaluation of volume and boundary integrals are avoided.…”

“…Suppose that an approximation to the double layer potential density φ(p) is known as an intermediate result of the simple iteration (22) or the GMRES iteration for the boundary integral equation (19) or (24). Denote the approximate density by φ υ ≡ φ ν (p).…”

“…The resulting scheme is of second-order accuracy and preserves the jumps across the interface. Modern versions of the II method are robust and efficient [22][23][24][25][26][27], and has been successfully applied to a variety of interfacerelated problems [28][29][30][31][32][33].…”