Superconvergence of the Gradient in Piecewise Linear Finite-Element Approximation to a Parabolic Problem

Abstract

Some recent results concerning maximum-norm superconvergence of the gradient in piecewise linear finite-element approximations of an elliptic problem are carried over to a parabolic problem. Both the standard semidiscrete in space Galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations, and the Crank-Nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes.

A posteriori error estimates by recovered gradients in parabolic finite element equations

“…Although Theorem 2.1 does not cover the Crank-Nicolson method, standard arguments, (cf. Theorem 2.3 in Thomée, Xu, and Zhang [13]), can be adapted to our problem up to dimension 2. Since the time discretization is of second order, we see good results already with 40 time steps.…”

Superconvergence of finite element method for parabolic problem

“…Superconvergence of finite element for parabolic problem has been studied by many authors. For example, Thomeé [8], Chen and Huang [1] studied superconvergence of the gradient in L 2 norm while Thomeé et al [9] studied maximum norm superconvergence of gradient for linear finite element. Superconvergence of the lumped finite element method for linear and nonlinear parabolic problems were studied in [2] and [6], respectively.…”

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

“…In two space dimensions, semidiscrete error estimates were studied in [3] and the fully discrete CrankNicolson method was studied in [33]. Since the main motivation of both investigations was the question of superconvergence of the gradient of the error, it was assumed that the solution is sufficiently smooth.…”

“…In both publications dealing with fully discrete error estimates, [16] and [33], the proofs are based on the splitting u − u kh = (u − R h u) + (R h u − u kh ), where R h is the Ritz projection. This idea was first introduced by M. Wheeler [34] in order to obtain optimal order error estimates in L 2 norm in space.…”