A Quasi-Projection Analysis of Galerkin Methods for Parabolic and Hyperbolic Equations

Superconvergence phenomena are demonstrated for Galerkin approxima- tions of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions. 1. Introduction. We shall be concerned primarily with the analysis of supercon- vergence phenomena associated with the numerical solution of second order, linear parabolic and hyperbolic equations by Galerkin methods based on piecewise-polyno- mial spaces. Our principal tool will be an asymptotic expansion to high order of the Galerkin solution; this expansion will be obtained by using a sequence of elliptic pro- jections and will be called a quasi-projection. In Sections 4 and 5 we develop the quasi-projection for parabolic Galerkin procedures for problems in one or several space variables for both Neumann and Dirichlet boundary conditions and derive optimal order negative norm estimates for the error in the Galerkin solution. In Section 6 we apply the quasi-projection to de- rive superconvergence results in the case of a single space variable when the Galerkin space consists of piecewise-polynomial functions of degree r. It is well known (4), (6), (7), (9), (10) that, if h is the knot spacing parameter associated with the not necessarily uniform grid, the Galerkin solution for standard parabolic problems con- verges with an error that is at best globally of order 0(hr+ ), as measured in L2 or L°°. Consider a knot at which the smoothness constraint of the Galerkin space re- duces to continuity. We show that the Galerkin solution produces an 0(h2r)-ap- proximation at such a knot. Also, we show that a very simply evaluated weighted quadrature of the Galerkin solution gives an cXft2^-approximation of the flux at the knot; the direct evaluation of the derivative of the Galerkin solution leads to an 0(/1r)-approximation. We summarize briefly in Section 7 results presented in detail elsewhere (3) showing that the superconvergence results above are preserved and that supercon- vergence occurs in the time increment when the Galerkin procedure is discretized in time by a collocation method. In Section 8 we treat continuous-time Galerkin methods for hyperbolic prob- lems and obtain analogous results. Throughout this paper we rely heavily on some earlier results of two of the