New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank-Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.

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.

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The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor w. A theory for determining w was given by Young ("Iterative methods for solving partial differential equations of elliptic types," Trans. Amer. Math. Soc., 76(1954), pp. 92-111) for consistently ordered matrices.Here we determine the optimal to for the 9-point stencil for the model problem of Laplace's equation on a square. We consider several orderings of the equations, including the natural rowwise and multicolor orderings, all of which lead to nonconsistently ordered matrices, and find two equivalence classes of orderings with different convergence behavior and optimal to's. We compare our results for the natural rowwise ordering to those of Garabedian ("Estimation of the relaxation factor for small mesh size," Math. Comp., 10 (1956), pp. 183-185) and explain why both results are, in a sense, correct, even though they differ. We also analyze a pseudo-SOR method for the model problem and show that it is not as effective as the SOR methods. Finally, we compare the point SOR methods to known results for line SOR methods for this problem.

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