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

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 $r\geq 1$ are used, which improve upon earlier results of Axelsson [Numer. Math. 28 (1977), pp. 1-14] requiring for 2d $r\geq 2$ and for 3d $r\geq 3.$ Based on quasi-projection technique introduced by Douglas {\it et al.} [Math. Comp.32 (1978),pp. 345-362], superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, {\it 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.