Circulant preconditioners for ill-conditioned boundary integral equations from potential equations. (English) Zbl 0932.65122
The authors study preconditioning of the Galerkin discretization of Symm’s integral equation on a smooth closed curve. They construct a circulant integral operator minimizing the distance to Symm’s integral operator with respect to the Hilbert-Schmidt norm. The discretization of the circulant integral gives the circulant preconditioner used for preconditioned conjugate gradient solving of the linear system. The spectral equivalence is proved and two numerical examples are given.
Reviewer: G.Schmidt (Berlin)
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 31A10 | Integral representations, integral operators, integral equations methods in two dimensions |
| 31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 65F10 | Iterative numerical methods for linear systems |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 65R20 | Numerical methods for integral equations |
Keywords:
boundary integral equations; first kind Fredholm equations; circulant preconditioner; preconditioned conjugate gradient method; Galerkin method; preconditioning; Symm’s integral equation; numerical examplesReferences:
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