Journal of Scientific Computing

, Volume 46, Issue 1, pp 71–99

An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations

Article

DOI: 10.1007/s10915-010-9384-z

Cite this article as:
Pani, A.K. & Yadav, S. J Sci Comput (2011) 46: 71. doi:10.1007/s10915-010-9384-z

Abstract

In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.

Keywords

LDG methodParabolic integro-differential equationSemidiscreteMixed type Ritz-Volterra projectionNegative norm estimatesRole of stabilizing parametersOptimal error bounds

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Industrial Mathematics GroupIndian Institute of Technology BombayMumbaiIndia