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OPTIMAL $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}L^2$ ESTIMATES FOR THE SEMIDISCRETE GALERKIN METHOD APPLIED TO PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Parabolic equations and systems

Published online by Cambridge University Press:  05 June 2014

DEEPJYOTI GOSWAMI ,
AMIYA K. PANI  and
SANGITA YADAV
DEEPJYOTI GOSWAMI
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam Tezpur 784028, Assam, India email deepjyoti@tezu.ernet.in
AMIYA K. PANI*
Affiliation:
Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India email akp@math.iitb.ac.in
SANGITA YADAV
Affiliation:
Department of Mathematics, Birla Institute of Technology and Science, Pilani, Pilani Campus, Rajasthan 333031, India email sangita.yadav@pilani.bits-pilani.ac.in
*
✉akp@math.iitb.ac.in
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Abstract

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We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal $L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in $L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.

Keywords

parabolic integro-differential equationfinite element methodsemidiscrete solutionenergy argumentoptimal error estimatenonsmooth initial datasuperconvergencemaximum norm estimate

MSC classification

Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 65M15: Error bounds 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 3 , January 2014 , pp. 245 - 266
Copyright
Copyright © 2014 Australian Mathematical Society 

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