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

TL;DR

This paper develops improved error estimates and superconvergence results for the Galerkin finite element method applied to strongly nonlinear parabolic problems, including negative norm estimates and nodal superconvergence in one dimension.

Contribution

It introduces new optimal error bounds using elliptic projection and quasi-projection techniques, extending superconvergence results and negative norm estimates for nonlinear parabolic problems.

Findings

Optimal error estimates for polynomial degree r≥1.

Superconvergence between Galerkin approximation and quasi-projection.

Negative norm a priori error estimates and nodal superconvergence in 1D.

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 $r\geq 1$ 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.