Volume 26, Issue 1
Error Analysis for a Fast Numerical Method to a Boundary Integral Equation of the First Kind

Jingtang Ma & Tao Tang

DOI:

J. Comp. Math., 26 (2008), pp. 56-68

Published online: 2008-02

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  • Abstract

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [{\em J. Comput. Math.}, 22 (2004), pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.

  • Keywords

Boundary integral equation Collocation method Graded mesh

  • AMS Subject Headings

65R20 45L10.

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COPYRIGHT: © Global Science Press

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@Article{JCM-26-56, author = {}, title = {Error Analysis for a Fast Numerical Method to a Boundary Integral Equation of the First Kind}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {1}, pages = {56--68}, abstract = { For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [{\em J. Comput. Math.}, 22 (2004), pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8610.html} }
TY - JOUR T1 - Error Analysis for a Fast Numerical Method to a Boundary Integral Equation of the First Kind JO - Journal of Computational Mathematics VL - 1 SP - 56 EP - 68 PY - 2008 DA - 2008/02 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8610.html KW - Boundary integral equation KW - Collocation method KW - Graded mesh AB - For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [{\em J. Comput. Math.}, 22 (2004), pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.
Jingtang Ma & Tao Tang . (1970). Error Analysis for a Fast Numerical Method to a Boundary Integral Equation of the First Kind. Journal of Computational Mathematics. 26 (1). 56-68. doi:
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