Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data

The purpose of this paper is to study the effect of numerical quadrature in the finite element analysis for a time dependent parabolic equation with nonsmooth initial data. Both semidiscrete and fully discrete schemes are analyzed using standard energy techniques. For the semidiscrete case, optimal order error estimates are derived in the L2 and H1-norms and quasi-optimal order in the L∞-norm, when the initial function is only in H01. Finally, based on the backward Euler method, a time discretization scheme is discussed and almost optimal rates of convergence in the L2, H1 and L∞-norms are established. © Springer-Verlag 1998.