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

Reddy, G. Murali; Sinha, Rajen

doi:10.1090/mcom/3067

On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations
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by G. Murali Mohan Reddy and Rajen K. Sinha;

Math. Comp. 85 (2016), 2365-2390

DOI: https://doi.org/10.1090/mcom/3067

Published electronically: December 30, 2015

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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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