Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems
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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 [Numer. Math. 28 (1977), pp. 1-14] requiring for 2d r≥ 2 and for 3d r≥ 3. Based on quasi-projection technique introduced by Douglas 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, 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.
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