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

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

References

Bramble, J. H., Pasciak, J. E. and Schatz, A. H., “The construction of preconditioners for elliptic problems by substructuring. I”, Math. Comp. 47 (1986) 103–134; doi:10.2307/2008084.Google Scholar
Cannon, J. R. and Lin, Y., “Nonclassical $H^1$ projection and Galerkin methods for nonlinear parabolic integro-differential equations”, Calcolo 25 (1988) 187–201; doi:10.1007/BF02575943.Google Scholar
Cannon, J. R. and Lin, Y., “A priori $L^2$ error estimates for finite-element methods for nonlinear diffusion equations with memory”, SIAM J. Numer. Anal. 27 (1990) 595–607 ; doi:10.1137/0727036.CrossRefGoogle Scholar
Cushman, J. H. and Ginn, T. R., “Nonlocal dispersion in media with continuously evolving scales of heterogeneity”, Transp. Porous Media 13 (1993) 123–138; doi:10.1007/BF00613273.Google Scholar
Dagan, G., “The significance of heterogeneity of evolving scales to transport in porous formations”, Water Resour. Res. 30 (1994) 3327–3336; doi:10.1029/94WR01798.CrossRefGoogle Scholar
Goswami, D. and Pani, A. K., “An alternate approach to optimal $L^2$-error analysis of semidiscrete Galerkin methods for linear parabolic problems with nonsmooth initial data”, Numer. Funct. Anal. Optim. 32 (2011) 946–982; doi:10.1080/01630563.2011.587334.CrossRefGoogle Scholar
Goswami, D., Pani, A. K. and Yadav, S., Optimal $L^2$-estimates for semi-discrete Galerkin methods for parabolic integro-differential equations with non-smooth data”, Report No. 09/38, Oxford University, 2009, available at http://eprints.maths.ox.ac.uk/858/1/finalOR38.pdf.Google Scholar
Huang, M. and Thomée, V., “Some convergence estimates for semidiscrete type schemes for time-dependent nonselfadjoint parabolic equations”, Math. Comp. 37 (1981) 327–346 ; doi:10.2307/2007430.Google Scholar
Lin, Y. P., “On maximum norm estimates for Ritz–Volterra projection with applications to some time dependent problems”, J. Comput. Math. 15 (1997) 159–178 , available at http://www.jcm.ac.cn/EN/Y1997/v15/12/159.Google Scholar
Lin, Y. P., Thomée, V. and Wahlbin, L. B., “Ritz–Volterra projections to finite-element spaces and applications to integrodifferential and related equations”, SIAM J. Numer. Anal. 28 (1991) 1047–1070; doi:10.1137/0728056.Google Scholar
Lin, Y. P. and Zhang, T., “The stability of Ritz–Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type”, Appl. Math. 36 (1991) 123–133 , available at http://hdl.handle.net/10338.dmlcz/104449.Google Scholar
Luskin, M. and Rannacher, R., “On the smoothing property of the Galerkin method for parabolic equations”, SIAM J. Numer. Anal. 19 (1982) 93–113; doi:10.1137/0719003.Google Scholar
Pani, A. K. and Peterson, T. E., “Finite element methods with numerical quadrature for parabolic integrodifferential equations”, SIAM J. Numer. Anal. 33 (1996) 1084–1105; doi:10.1137/0733053.Google Scholar
Pani, A. K. and Sinha, R. K., “Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data”, Calcolo 35 (1998) 225–248; doi:10.1007/s100920050018.CrossRefGoogle Scholar
Pani, A. K. and Sinha, R. K., “On the backward Euler method for time dependent parabolic integro-differential equations with nonsmooth initial data”, J. Integral Equations Appl. 10 (1998) 219–249; doi:10.1216/jiea/1181074222.Google Scholar
Pani, A. K. and Sinha, R. K., “Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data”, Calcolo 37 (2000) 181–205; doi:10.1007/s100920070001.Google Scholar
Pani, A. K. and Sinha, R. K., “Finite element approximation with quadrature to a time dependent parabolic integro-differential equation with nonsmooth initial data”, J. Integral Equations Appl. 13 (2001) 35–72; doi:10.1216/jiea/996986882.CrossRefGoogle Scholar
Pani, A. K., Thomée, V. and Wahlbin, L. B., “Numerical methods for hyperbolic and parabolic integro-differential equations”, J. Integral Equtions Appl. 4 (1992) 533–584 ; doi:10.1216/jiea/1181075713.Google Scholar
Renardy, M., Hrusa, W. J. and Nohel, J. A., Mathematical problems in viscoelasticity, Volume 35 of Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman Scientific & Technical, Harlow, 1987).Google Scholar
Thomée, V., Galerkin finite element methods for parabolic problems, 2nd edn. Volume 25 of Springer Series in Computational Mathematics (Springer, Berlin, 2006).Google Scholar
Thomée, V. and Zhang, N.-Y., “Error estimates for semidiscrete finite element methods for parabolic integro-differential equations”, Math. Comp. 53 (1989) 121–139; doi:10.2307/2008352.CrossRefGoogle Scholar
Thomée, V. and Zhang, N., “Backward Euler type methods for parabolic integro-differential equations with nonsmooth data”, in: Contributions in numerical mathematics, Volume 2 of World Scientific Series in Applicable Analysis (World Scientific, River Edge, NJ, 1993), 373–388; doi:10.1142/9789812798886_0029.Google Scholar
Yanik, E. G. and Fairweather, G., “Finite element methods for parabolic and hyperbolic partial integro-differential equations”, Nonlinear Anal. 12 (1988) 785–809 ; doi:10.1016/0362-546X(88)90039-9.Google Scholar