This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems.
Abstract. This work presents techniques, theory, and numbers for multigrid in a general ddimensional setting. The main focus of this paper is the multigrid convergence for high-dimensional partial differential equations on non-equidistant grids such as may be encountered in a sparse-grid solution. As a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube. We present some techniques for building the general d-dimensional adaptations of the multigrid components and propose grid-coarsening strategies to handle anisotropies that are induced due to discretization on a non-equidistant grid. Apart from the practical formulas and techniques, we present-in detail-the smoothing analysis of the point ω-red-black Jacobi method for a general multidimensional case. We show how relaxation parameters may be evaluated efficiently and used for better convergence. This analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second-and the fourth-order finite difference operators. Finally we present some results derived from numerical experiments based on the test problem.Key words. multigrid, high-dimensional PDEs, anisotropic diffusion equation, coarsening strategies, point-smoothing methods, relaxation parameters, Fourier-smoothing analysis AMS subject classifications. 65N55, 65F10, 65Y20 DOI. 10.1137/060665695 Introduction. Multidimensional partial differential equations have diverse applications in various fields of applied sciences, including financial engineering [8], molecular biology [3], and quantum dynamics [1,21]. There are quite a few fast and efficient solution techniques for partial differential equations (henceforth PDEs) of which multigrid ranks among the best. Multigrid is a well-known iterative procedure for the solution of large and sparse linear systems that arise from various kinds of PDE discretizations. The existing literature on the multigrid treatment of various problems, however, rarely explores issues that arise out of growth in the dimensionality of the problem. The implications of dimensionality growth include deterioration of the multigrid convergence rate, impractical storage requirements, and huge amounts of CPU time for single grid solution methods. Our main emphasis in this paper lies on the first challenge. We abbreviate multigrid for d-dimensional PDEs as d-multigrid. In this paper d represents both abstract dimensionality and dimensions. So, e.g., 3d is to be interpreted as 3-dimensional.The major hinderance in the numerical solution of multidimensional PDEs is the so-called curse of dimensionality, which implies that with the growth in dimensions we have an exponential growth in the number of grid points. This increases the
Abstract. The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.
In this work, we present a geometric multigrid method for PDEs discretized on stretched grids. The emphasis is on geometric L‐shaped coarsening techniques that we have developed in this context. The presented method is matrix free, in contrast with alternatives such as algebraic multigrid or certain preconditioned Krylov‐subspace‐based solution methods. For a Poisson model problem, we explain, both visually and in a descriptive way, how the stretched fine grid may yield a sequence of coarser grids so as to maintain the complementarity between relaxation and coarse grid correction. We also present complexity estimates of the method, thus demonstrating its efficiency. Through figures and numerical experiment tables, we provide convergence histories for the model problem discretized and solved on various stretched grids with our method. Copyright © 2009 John Wiley & Sons, Ltd.
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