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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Discontinuous Galerkin method for an evolution equation with a memory term of positive type

Authors: Kassem Mustapha and William McLean
Journal: Math. Comp. 78 (2009), 1975-1995
MSC (2000): Primary 26A33, 45J05, 65M12, 65M15, 65M60
Published electronically: February 23, 2009
MathSciNet review: 2521275
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Abstract: We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by $ C(t-s)^{\alpha-1}$, where $ 0<\alpha<1$. For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most $ q-1$, for $ q=1$ or $ 2$. For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order $ k^q+h^2\ell(k)$, where $ k$ and $ h$ are the mesh sizes in time and space, respectively, and $ \ell(k)=\max(1,\log k^{-1})$. In the case $ q=2$, we prove a higher convergence rate of order $ k^3+h^2\ell(k)$ at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at $ t=0$, necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.

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Additional Information

Kassem Mustapha
Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

William McLean
Affiliation: School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia

Keywords: Memory term, discontinuous Galerkin method, a priori error estimates, non-uniform time steps, finite element method
Received by editor(s): October 16, 2007
Received by editor(s) in revised form: October 9, 2008
Published electronically: February 23, 2009
Additional Notes: Support of the KFUPM is gratefully acknowledged
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.