On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem

• Published in: Journal of computational and applied mathematics Vol. 156; no. 2; pp. 303 - 318
• Main Authors: Harris, Paul J., Chen, Ke
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.07.2003; Elsevier
• Subjects: Boundary integral equation; Burton–Miller; CGN; Exact sciences and technology; Exterior Helmholtz; Galerkin; GMRES; Hyper-singular operators; Integral equations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Partial differential equations; Preconditioners; Sciences and techniques of general use; Exterior Helmholtz; Boundary integral equation; GMRES; Hyper-singular operators; Preconditioners; Burton–Miller; CGN; Galerkin; Integral operator; Fast solver; Convergence of numerical methods; Three-dimensional calculations; Fredholm operator; Singular equation; Iterative method; Numerical method; Boundary condition; Partial differential equation; Numerical approximation; Quadrature; Singular integral; Approximation order; Integral; Integral equation; Collocation method; Gradient method; Boundary element method; Conjugate gradient method; Neumann problem; 2001; Helmholtz equation; Algorithm; Green function; Galerkin method; Preconditioning; Singular operator
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(02)00918-4

The paper presents a Galerkin numerical method for solving the hyper-singular boundary integral equations for the exterior Helmholtz problem in three dimensions with a Neumann's boundary condition. Previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and Galerkin methods are not available due to the presence of a hypersingular integral operator. This paper proposes a high order Galerkin method by using singularity subtraction technique to reduce the hyper-singular operator to a weakly singular one. Moreover, we show here how to extend the previous work (J. Appl. Numer. Math. 36 (4) (2001) 475–489) on sparse preconditioners to the Galerkin case leading to fast convergence of two iterative solvers: the conjugate gradient normal method and the generalised minimal residual method. A comparison with the collocation method is also presented for the Helmholtz problem with several wavenumbers.