Direct methods for sparse matrices
Direct methods for sparse matrices
How fast are nonsymmetric matrix iterations
SIAM Journal on Matrix Analysis and Applications
Preconditioning for boundary integral equations
SIAM Journal on Matrix Analysis and Applications
A boundary element method for the Helmholtz equation using finite part integration
Computer Methods in Applied Mechanics and Engineering
Integral equations: theory and numerical treatment
Integral equations: theory and numerical treatment
Adaptively Preconditioned GMRES Algorithms
SIAM Journal on Scientific Computing
On a Class of Preconditioning Methods for Dense Linear Systems from Boundary Elements
SIAM Journal on Scientific Computing
An Analysis of Sparse Approximate Inverse Preconditioners for Boundary Integral Equations
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
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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.