Resumen de B-spline approximation of elliptic problems with non-smooth coefficients
Weighted B-splines approximate smooth solutions of elliptic problems with maximal 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 −∇ · (α∇u) = f in Ω, u = 0 on ∂Ω, where α is discontinuous across a submanifold Γ ⊂ Ω. We show for the twodimensional 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 Γ 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.
