DELFT UNIVERSITY OF TECHNOLOGY REPORT 02-12 Some Numerical Aspects for Solving Sparse Large Linear Systems Derived from the Helmholtz Equation

In this report, several numerical aspects and difficulties for solving a linear system derived from the time-harmonic wave equations are overviewed. The presentation begins with the derivation of the governing equation for waves propagating in general inhomogeneous media. Due to the need of numerical solutions, various discretizations based on finite difference are discussed. Some numerical methods which are considered applicable for solving the resulting large but sparse, indefinite, non-Hermitian linear system are discussed, including some types of existing preconditioners to possibly accelerate the convergence. Following demands for solving large linear problems in parallel computers, domain decomposition methods are revisited, with particular attentions on methods for solving the discrete Helmholtz equation.