An efficient multigrid algorithm for heterogeneous acoustic media sign‐indefinite high‐order FEM models

Abstract

Summary Large‐scale scientific computing models are needed for the simulation of wave propagation especially for multiple frequency and high‐frequency models in complex heterogeneous media. Multigrid methods provide efficient iterative solvers for many large sign‐definite systems of equations resulting from physical models. Time‐harmonic wave propagation models lead to sign‐indefinite systems with eigenvalues in the left half of the complex plane. Thus standard multigrid approaches applied in conjunction with a low‐order finite difference or finite element method are not sufficient. In this work, we describe a high‐order finite element method model for multiple (low to high) frequency time‐harmonic acoustic wave propagation on general curved, non‐convex, and non‐smooth domains with heterogeneous media using a multigrid approximation of the shifted Laplacian operator as a preconditioner. We implement the model using an efficient geometric multigrid approach with parallel grid transfer operator calculations to simulate the model using the BiCGStab iterative solver. We demonstrate the efficiency and parallel performance of the computational model with multiple low (5 wavelength) to high‐frequency (100 wavelength) input incident waves. Copyright © 2016 John Wiley & Sons, Ltd.

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

“…The exterior wave propagation BEM models lead to dense complex algebraic systems, and the standard variational formulation based interior wave FEM models lead to sparse complex systems with their eigenvalues in the left half of the complex plane [26,38]. Developing efficient preconditioned iterative solvers for such systems has also dominated research activities over the last two decades [19], in conjunction with efficient implementations using multigrid and domain decomposition techniques, see [25,27] and references therein.…”

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

“…[13][14][15] In recent years, there has been a great effort to develop efficient solvers for systems arising from (1), using several different approaches to tackle the problem. One of the most common approaches is the shifted Laplacian multigrid preconditioner, 12,13,[16][17][18][19][20][21][22] which modifies the equation by adding complex values to the diagonal of the matrix. The modified system is then solved using a multigrid method and is used as a preconditioner for the nonshifted system.…”