Rapid solution of integral equations of classical potential theory.

*(English)*Zbl 0629.65122A 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.

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.) |

##### Keywords:

boundary integral equation method; comparison of methods; Laplace equation; single or double layer potential; Nyström algorithm; trapezoidal quadrature rule; conjugate residual algorithm; harmonic expansions; fast Poisson solvers
Full Text:
DOI

##### References:

[1] | Atkinson, K.E., A survey of numerical methods for the solution of Fredholm integral equations of the second kind, (1976), SIAM Philadelphia, Pa · Zbl 0353.65069 |

[2] | Blue, J.L., Boundary integral solutions of Laplace’s equation, () · Zbl 0392.35014 |

[3] | Churchill, R.V., Complex variables and applications, (1960), McGraw-Hill New York · Zbl 0092.06802 |

[4] | Courant, R.; Hilbert, D., Methods of mathematical physics, (1966), Interscience New York · Zbl 0729.00007 |

[5] | Duff, I.S., Recent developments in the solution of large sparse linear equations, () · Zbl 0437.65024 |

[6] | Eisenstat, S.C.; Elman, H.C.; Schultz, M.H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. numer. anal., 345-357, (1983) · Zbl 0524.65019 |

[7] | () |

[8] | Hess, J.K., A higher order panel method for three-dimensional potential flow, Douglas aircraft co., report no. MDC J8519, (1979) |

[9] | Hess, J.L.; Smith, A.M.O., Calculation of potential flow about arbitrary bodies, Prog. aero. sci., 8, 1-138, (1967) · Zbl 0204.25602 |

[10] | Kantorovich, L.V.; Krylov, V.I., Approximate methods of higher analysis, (1964), Wiley New York · Zbl 0040.21503 |

[11] | Koshliakov, N.S.; Smirnov, M.M.; Gliner, E.B., Differential equations of mathematical physics, (1964), North-Holland Amsterdam · Zbl 0115.30701 |

[12] | Mayo, A., The fast solution of Poisson’s and biharmonic equations in irregular regions, SIAM J. numer. anal., 21, No. 2, (1984) · Zbl 1131.65303 |

[13] | Petrovsky, I.G., Lectures on partial differential equations, (1954), Interscience London · Zbl 0059.08402 |

[14] | Rokhlin, V., Solution of acoustic scattering problems by means of second kind integral equations, Wave motion, 5, 257-272, (1983) · Zbl 0522.73022 |

[15] | Schwartztrauber, P.N.; Sweet, R.A., Algorithm 541, ACM trans. math. software, 5, 352, (1979) |

[16] | Stepleman, R.S., Some refinements of method of moments for the solution of the charged lamina problems in 3-D, (), 34-40 |

[17] | {\scR. S. Stepleman}, personal communication, 1981. |

[18] | Stoer, J.; Bulirsch, R., Introduction to numerical analysis, (1980), Springer-Verlag New York · Zbl 0423.65002 |

[19] | Winthner, R., Some superlinear convergence results for the conjugate gradient method, SIAM J. numer. anal., 17, No. 1, (1981) |

[20] | Wouk, A., A course of applied functional analysis, (1979), Wiley New York · Zbl 0407.46001 |

[21] | {\scD. P. Young}et al., SIAM J. Statist. Sci. Comput., in press. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.