Spline qualocation methods for boundary integral equations (Q909427)

This paper further develops the qualocation method for the solution of integral equations on smooth closed curves. Qualocation is a (Petrov-) Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Here we allow the use of splines as trial and test functions, and prove stability and convergence results for certain qualocation rules when the underlying (Petrov-) Galerkin method is stable. In the common case of a first kind equation with the logarithmic kernel, a typical qualocation method employs piecewise-linear splines, and performs the outer integrals using just two points per interval. This method, with appropriate choice of the quadrature points and weights, has order \(O(h^ 5)\) convergence in a suitable negative norm. Numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions.