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Rapid solution of integral equations of classical potential theory. (English) Zbl 0629.65122
A rapid solution of integral equations is described which is applicable to Dirichlet and Neumann boundary value problems for the Laplace equation. The method has computational requirements proportional to n, where n is the number of nodes on the boundary. It uses the classical approach of transforming the problem to an integral equation for the single or double layer potential on the boundary. This equation is then discretized using the Nyström algorithm associated with the trapezoidal quadrature rule. The resulting system is solved by the generalized conjugate residual algorithm (GCRA). The decrease of computational requirements is achieved by reducing the number of operations needed for applying a matrix to a vector in the process of solving by the GCRA. This is made possible by approximations based on harmonic expansions.
The algorithm is tested on some standard problems which confirm the theoretically predicted properties. It must, however, be kept in mind that the method is superior to fast Poisson solvers only when the solution in a limited number of points outside the boundary is required.
Reviewer: P.Polcar

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35C15 Integral representations of solutions to PDEs
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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