SDouglas, Jim, Jr.;Dupont, Todd ;Wheeler, Mary Fanett ;MRC-TSR-1382DA-31-124-ARO(D)-462Sponsored in part by National Science Foundation.*Heat transfer, *Partial differential equations, Calculus of variations, Convergence, Approximation, Theorems*Galerkin method, Parabolic differential equations, Heat equationThomee and Wahlbin have introduced a Galerkin method for the heat equation in a single space variable based on the (H sup 1)-inner product and have obtained (H sup 2) and (H sup 1) estimates for the error. An (L sup 2) estimate is given here. The main object is to show knot superconvergence phenomena when the subspace is a piecewise-polynomial space. For (C sup 2)-piecewise-polynomials of degree r, the error in the knot values is O(h sup(2r-2)); for the (C sup 1) case, both knot values and knot first x-derivatives are approximated to within O(h sup(2r-2)). (Author).