Wavelet approximations for first kind boundary integral equations on polygons

Summary.

An elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation. Its Galerkin discretization with $N$ degrees of freedom on the boundary with spline wavelets as basis functions is analyzed. A truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from $O(N^2)$ to $O(N\log N)$ entries. The condition numbers are bounded independently of the meshwidth. It is proved that the compressed scheme thus obtained yields in $O(N(\log N)^2)$ operations approximate solutions with the same asymptotic convergence rates as the full Galerkin scheme in the boundary energy norm as well as in interior points. Numerical examples show the asymptotic error analysis to be valid already for moderate values of $N$.