On the Superconvergence of Galerkin Methods for Hyperbolic IBVP

Finite-element Galerkin methods using B-splines of order r for periodic first-order hyperbolic equations exhibit superconvergence on uniform grids (mesh size h) at the nodes; i.e., there is an error estimate O (hr). In this paper it will be shown that no matter how the approximating subspace Shis modified in a boundary layer [ 0, (s - 1)h], s arbitrary but fixed, the superconvergence property is lost for the hyperbolic model problem$u_t = u_x, 0 \leq\le x < \infty, t \geq 0$. We shall also discuss the implications of this result when constructing compact implicit difference schemes.