A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov-Galerkin method with quadrature, for solving second-order, variable coefficient, elliptic boundary value problems on rectangular domains. In our scheme, the trial space consists of C-2 splines of degree r greater than or equal to 3, the test space consists of C-0 splines of degree r - 2, and we use composite (r - 1)-point Gauss quadrature. We show existence and uniqueness of the approximate solution and establish optimal order error bounds in H-2, H-1 and L-2 norms.