Chapter

Numerical Analysis of Multiscale Problems

Volume 83 of the series Lecture Notes in Computational Science and Engineering pp 127-161

Date:

Fast Algorithms for High Frequency Wave Propagation

  • Björn EngquistAffiliated withDepartment of Mathematics and ICES, The University of Texas at Austin Email author 
  • , Lexing YingAffiliated withDepartment of Mathematics and ICES, The University of Texas at Austin

Abstract

High frequency wave propagation is computationally challenging due to the very large number of unknowns that are needed in direct numerical approximations. We will present new fast algorithms for the solution of the linear systems, which follow from discretization of the Helmholtz equation and its related integral equation formulation. For the Helmholtz equation we present a new type of preconditioner, which, together with the GMRES iterative method, results in a near optimal computational complexity. The cost of the preconditioner scales essentially linearly with the number of unknowns and the number of iterations is independent of frequency. In the integral equation case, a directional fast multilevel technique also results in a near optimal computational complexity.