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Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation
Author(s) -
Engquist Björn,
Ying Lexing
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20358
Subject(s) - preconditioner , helmholtz equation , mathematics , incomplete lu factorization , generalized minimal residual method , solver , discretization , factorization , coefficient matrix , matrix (chemical analysis) , representation (politics) , helmholtz free energy , iterative method , boundary value problem , mathematical analysis , matrix decomposition , mathematical optimization , algorithm , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , politics , political science , law
The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable‐coefficient Helmholtz equation including very‐high‐frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three‐dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc.