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Optimal order collocation for the mixed boundary value problem on polygons

Author: Pascal Laubin
Journal: Math. Comp. 70 (2001), 607-636
MSC (2000): Primary 65N35, 65R20; Secondary 45B05
Published electronically: March 2, 2000
MathSciNet review: 1813142
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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.

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Additional Information

Pascal Laubin
Affiliation: Université de Liège, Institut de Mathématique, Grande Traverse 12, B-4000 Liège, Belgium

Keywords: Collocation method, mixed problem, optimal order
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
Article copyright: © Copyright 2000 American Mathematical Society