The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

Abstract

Superconvergence in the L 2 {L^2} -norm for the Galerkin approximation of the integral equation L u = f Lu = f is studied, where L is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let u h {u_h} be the Galerkin approximation to u. By using the K-operator, an operator that averages the values of u h {u_h} , we will construct a better approximation than u h {u_h} itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm’s equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.