B-spline approximation of elliptic problems with non-smooth coefficients

Weighted B-splines approximate smooth solutions of elliptic problems with max-imal order. However, lack of regularity due to non-smooth coefficients of the partial differential equation can cause a severe loss of accuracy. A typical model problem is -& nabla; middot (alpha & nabla;u) = f in omega, u = 0 on & part;omega, where alpha is discontinuous across a submanifold gamma subset of omega. We show for the two-dimensional case, that the optimal convergence rates can be retained if we augment addition so-called singular splines to the weighted spline basis. The singular splines are constructed with an implicit representation of gamma and can model the discontinuous gradients of solutions accurately. As a result we obtain a meshless method of optimal order with the computational advantages of the B-spline calculus.