Optimal \(L_{2}\) estimates for semidiscrete Galerkin method applied to parabolic integro-differential equations with nonsmooth data

Amiya Kumar Pani, Deepjyoti Goswami, Sangita Yadav

Abstract


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.


doi:10.1017/S1446181114000030

Keywords


parabolic integro-differential equation, finite element method, semidiscrete solution, energy argument, optimal error estimate, nonsmooth initial data, superconvergence, maximum norm estimate



DOI: http://dx.doi.org/10.21914/anziamj.v55i0.5522



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.