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On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations


Authors: G. Murali Mohan Reddy and Rajen K. Sinha
Journal: Math. Comp. 85 (2016), 2365-2390
MSC (2010): Primary 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3067
Published electronically: December 30, 2015
MathSciNet review: 3511285
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Abstract: In this exposition, we derive two anisotropic error estimators for parabolic integro-differential equations in a two-dimensional convex polygonal domain. A continuous, piecewise linear finite element space is employed for the space discretization and the time discretization is based on the Crank-Nicolson method. The a posteriori contributions corresponding to space discretization is derived using the anisotropic interpolation estimates together with the Zienkiewicz-Zhu error estimator to approximate the error gradient. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to time discretization. Moreover, linear approximations of the Volterra integral term are used in a crucial way to estimate the quadrature error in the approximation of the Volterra integral term.


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Additional Information

G. Murali Mohan Reddy
Affiliation: Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India
Email: gmuralireddy1984@gmail.com

Rajen K. Sinha
Affiliation: Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India
Email: rajen@iitg.ernet.in

DOI: https://doi.org/10.1090/mcom/3067
Keywords: Parabolic integro-differential equations, finite element method, Crank-Nicolson scheme, anisotropic error estimator
Received by editor(s): December 8, 2013
Received by editor(s) in revised form: April 1, 2015
Published electronically: December 30, 2015
Additional Notes: The second author is the corresponding author
Article copyright: © Copyright 2015 American Mathematical Society