Superconvergence of a finite element method for linear integro-differential problems
type: Item
instance of: scholarly article
title: Superconvergence of a finite element method for linear integro-differential problems
zbMATH Open document ID: 0966.65118
DOI: 10.1155/S0161171200001940
author: Qian Li
author: Do Y. Kwak
author: Sungyun Lee
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2001-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/48850
review text: The authors introduce a new way of approximating the initial condition to the semidiscrete finite element method for the following problem with memory \[ u_t=\nabla\left\{a(x,t)\nabla u+ \int_0^tb(x,t,\tau)\nabla u(x, \tau) d\tau\right\}+f(x,t),\quad (x,t)\in\Omega\times (0,T], \] \[ u(x,0) =u_0(x),\quad x\in\Omega,\quad u(x,t)=0,\quad x\in\partial \Omega\times[0,T], \] where \(\Omega\in \mathbb R^2\) is a bounded domain with boundary \(\partial \Omega\), \(a, b, f, u_0\) are bounded together with their derivatives and \(0<a_0\leq a(x,t), (x, t)\in\Omega\times[0,T].\) Several superconvergence results are obtained for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. For \(k > 1,\) first order gain in \(L_p(2\leq p\leq\infty)\) norm, second order in \(W^{1,p}(2\leq p\leq\infty)\) norm and almost second order in \(W^{1,\infty}\) norm are obtained. For \(k=1,\) first order gain in \(W^{1,p}(2\leq p\leq\infty)\) norm is derived. Applying interpolated postprocessing technique to the approximate solution, one order global superconvergence between the exact solution and the interpolation of the approximate solution in the \(L_p\) and \(W^{1,p}(2\leq p\leq\infty)\) is obtained.
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 1485600
zbMATH Keywords: finite element method
zbMATH Keywords: Volterra integro-differential equation
zbMATH Keywords: Ritz-Volterra projection
zbMATH Keywords: semidiscrete Galerkin approximation scheme
zbMATH Keywords: superconvergence
zbMATH Keywords: postprocessing technique
zbMATH Keywords: error estimates
MaRDI profile type: Publication
OpenAlex ID: W2150762328
description: scientific article; zbMATH DE number 1485600