ON THE SUPERCONVERGENCE OF GALERKIN METHODS FOR HYPERBOLIC IBVP

Finite-element Galerkin methods using B-splines of order r for periodi c first-order hyperbolic equations exhibit superconvergence on uniform grids (mesh size h) at the nodes; i.e., there is an error estimate O( h(2r)) instead of the expected convergence rate O(h(r)). In this paper it will be shown that no matter how the approximating subspace S-h is modified in a boundary layer [0, (s - 1)h], s arbitrary but fixed, th e superconvergence property is lost for the hyperbolic model problem u (t) = u(x), 0 less than or equal to x < infinity, t greater than or eq ual to 0. We shall also discuss the implications of this result when c onstructing compact implicit difference schemes.