A multiscale method for the double layer potential equation on a polyhedron.
- Dahmen, Wolfgang
- Kleemann, Bernd
- Prößdorf, Siegfried
- Schneider, Reinhold
2010 Mathematics Subject Classification
- 65N38 65N35 65N30 31B10 35J05
- boundary element method, boundary integral equations, Laplace equation, exterior domain problem, unbounded domains, double layer potential equation, polyhedra; collocation, multiscale decompositions, linear finite element spaces, sparse matrices, algorithm, numerical experiments
This paper is concerned with the numerical solution of the double layer potential equation on polyhedra. Specifically, we consider collocation schemes based on multiscale decompositions of piecewise linear finite element spaces defined on polyhedra. An essential difficulty is that the resulting linear systems are not sparse. However, for uniform grids and periodic problems one can show that the use of multiscale bases gives rise to matrices that can be well approximated by sparse matrices in such a way that the solutions to the perturbed equations exhibits still sufficient accuracy. Our objective is to explore to what extent the presence of corners and edges in the domain as well as the lack of uniform discretizations affects the performance of such schemes. Here we propose a concrete algorithm, describe its ingredients, discuss some consequences, future perspectives, and open questions, and present the results of numerical experiments for several test domains including non-convex domains.
- Advances in Computational Mathematics (H.P. Dikshit and C.A. Micchelli, eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, pp. 15--57, 1994