Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems

Abstract

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r≥1 are used, which improve upon earlier results of Axelsson ((1977) [3]) requiring for 2d r≥2 and for 3d r≥3. Based on quasi-projection technique introduced by Douglas et al. ((1978) [11]), superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, a priori error estimates in Sobolev spaces of negative index are derived. Moreover, in a single space variable, nodal superconvergence results between the true solution and Galerkin approximation are established.