### Abstract

A boundary element method (BEM), without any numerical integration, is presented here for the treatment of boundary value problems in Laplace's equation on a plane domain with a polygonal boundary. We use the double-layer potential to approximate the solution. The exact forms of the double-layer potentials for the first- and second-order spline density are to be derived here. These potentials can be differentiated directly. Then, the collocation method is applied to mixed boundary value problems. The collocation equations of the first-order scheme for Dirichlet's problems on a convex domain are shown to be well-conditioned; i.e., the condition number of the resulting matrix is bounded by a constant independent of the number of the elements. The second-order method can be applied to mixed boundary problems on interior and exterior regions. Two mixed boundary value problems with singular solutions are solved by the second-order scheme. Numerical examples show that the scheme is capable of dealing with singularities.

Original language | English |
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Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Applied Mathematics and Computation |

Volume | 123 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 Sep 10 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

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**A boundary element method for Laplace's equation without numerical integrations.** / Shen, Shih-Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A boundary element method for Laplace's equation without numerical integrations

AU - Shen, Shih-Yu

PY - 2001/9/10

Y1 - 2001/9/10

N2 - A boundary element method (BEM), without any numerical integration, is presented here for the treatment of boundary value problems in Laplace's equation on a plane domain with a polygonal boundary. We use the double-layer potential to approximate the solution. The exact forms of the double-layer potentials for the first- and second-order spline density are to be derived here. These potentials can be differentiated directly. Then, the collocation method is applied to mixed boundary value problems. The collocation equations of the first-order scheme for Dirichlet's problems on a convex domain are shown to be well-conditioned; i.e., the condition number of the resulting matrix is bounded by a constant independent of the number of the elements. The second-order method can be applied to mixed boundary problems on interior and exterior regions. Two mixed boundary value problems with singular solutions are solved by the second-order scheme. Numerical examples show that the scheme is capable of dealing with singularities.

AB - A boundary element method (BEM), without any numerical integration, is presented here for the treatment of boundary value problems in Laplace's equation on a plane domain with a polygonal boundary. We use the double-layer potential to approximate the solution. The exact forms of the double-layer potentials for the first- and second-order spline density are to be derived here. These potentials can be differentiated directly. Then, the collocation method is applied to mixed boundary value problems. The collocation equations of the first-order scheme for Dirichlet's problems on a convex domain are shown to be well-conditioned; i.e., the condition number of the resulting matrix is bounded by a constant independent of the number of the elements. The second-order method can be applied to mixed boundary problems on interior and exterior regions. Two mixed boundary value problems with singular solutions are solved by the second-order scheme. Numerical examples show that the scheme is capable of dealing with singularities.

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U2 - 10.1016/S0096-3003(00)00009-6

DO - 10.1016/S0096-3003(00)00009-6

M3 - Article

VL - 123

SP - 1

EP - 25

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 1

ER -