## Optimal order collocation for the mixed boundary value problem on polygons

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**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