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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Optimal order collocation for the mixed boundary value problem on polygons
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by Pascal Laubin PDF
Math. Comp. 70 (2001), 607-636 Request permission

Abstract:

In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not $1$. This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order $2m-1$ are used, we get quasi-optimal estimates in $H^m$-norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.
References
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Additional Information
  • Pascal Laubin
  • Affiliation: Université de Liège, Institut de Mathématique, Grande Traverse 12, B-4000 Liège, Belgium
  • Email: P.Laubin@ulg.ac.be
  • Received by editor(s): May 22, 1998
  • Received by editor(s) in revised form: November 17, 1998, and March 16, 1999
  • Published electronically: March 2, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 607-636
  • MSC (2000): Primary 65N35, 65R20; Secondary 45B05
  • DOI: https://doi.org/10.1090/S0025-5718-00-01209-6
  • MathSciNet review: 1813142