The tolerant qualocation method for variable-coefficient elliptic equations on curves

Abstract

The 'tolerant' modification of the qualocation method is studied for variable-coefficient elliptic equations on curves. The modification (in which the discrete innerproducts on the righthand side of the qualocation method are replaced by exact integration) allows the same high-order convergence as the standard spline qualocation method but with reduced smoothness assumptions on the exact solution. The study (improving upon previous work for constant-coefficient boundary integral equations) builds upon a recent extension of the standard qualocation method to equations with variable coefficients by Sloan and Wendland. In particular, it is shown that, with exactly the same 'qualocation' rules as in that recent work for the standard qualocation method, the tolerant version of the method achieves the full order of convergence of the standard method but with just the same smoothness assumption on the exact solution as in the Galerkin method. The tolerant version of the method therefore allows convergence of arbitrarily high order to be achieved (in appropriate negative norms, and for splines of high enough order) even when the exact solution is not smooth.