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

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006), no. 254, 511–531. MR 2196979, DOI 10.1090/S0025-5718-05-01800-4
E. Bänsch, F. Karakatsani, and Ch. Makridakis, A posteriori error control for fully discrete Crank-Nicolson schemes, SIAM J. Numer. Anal. 50 (2012), no. 6, 2845–2872. MR 3022245, DOI 10.1137/110839424
V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl. 103 (1984), no. 2, 575–588. MR 762575, DOI 10.1016/0022-247X(84)90147-1
L. Formaggia and S. Perotto, New anisotropic a priori error estimates, Numer. Math. 89 (2001), no. 4, 641–667. MR 1865506, DOI 10.1007/s002110100273
L. Formaggia and S. Perotto, Anisotropic error estimates for elliptic problems, Numer. Math. 94 (2003), no. 1, 67–92. MR 1971213, DOI 10.1007/s00211-002-0415-z
Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI 10.1007/BF00281373
G. J. Habetler and R. L. Schiffman, A finite difference method for analyzing the compression of poro-viscoelastic media, Computing (Arch. Elektron. Rechnen) 6 (1970), 342–348 (English, with German summary). MR 286371, DOI 10.1007/bf02238819
W. E. Kastenberg and P. L. Chambre, On the stability of nonlinear space dependent reactor kinetics, Nucl. Sci. Eng., 31 (1968), pp. 67–79.
Alexei Lozinski, Marco Picasso, and Virabouth Prachittham, An anisotropic error estimator for the Crank-Nicolson method: application to a parabolic problem, SIAM J. Sci. Comput. 31 (2009), no. 4, 2757–2783. MR 2520298, DOI 10.1137/080715135
Stefano Micheletti and Simona Perotto, Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 9-12, 799–835. MR 2195291, DOI 10.1016/j.cma.2005.02.009
M. Picasso, An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems, SIAM J. Sci. Comput. 24 (2003), no. 4, 1328–1355. MR 1976219, DOI 10.1137/S1064827501398578
M. Picasso, Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator, Comm. Numer. Methods Engrg. 19 (2003), no. 1, 13–23. MR 1952014, DOI 10.1002/cnm.546
M. Picasso, Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives, Comput. Methods Appl. Mech. Engrg. 196 (2006), no. 1-3, 14–23. MR 2270123, DOI 10.1016/j.cma.2005.11.018
Marco Picasso and Virabouth Prachittham, An adaptive algorithm for the Crank-Nicolson scheme applied to a time-dependent convection-diffusion problem, J. Comput. Appl. Math. 233 (2009), no. 4, 1139–1154. MR 2557303, DOI 10.1016/j.cam.2009.09.004
G. Murali Mohan Reddy and Rajen K. Sinha, Ritz-Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations, IMA J. Numer. Anal. 35 (2015), no. 1, 341–371. MR 3335208, DOI 10.1093/imanum/drt059
Simon Shaw and J. R. Whiteman, Numerical solution of linear quasistatic hereditary viscoelasticity problems, SIAM J. Numer. Anal. 38 (2000), no. 1, 80–97. MR 1770343, DOI 10.1137/S0036142998337855
Simon Shaw and J. R. Whiteman, A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems, Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 52, 5551–5572. MR 2103153, DOI 10.1016/j.cma.2004.04.005
Vidar Thomée and Nai Ying Zhang, Error estimates for semidiscrete finite element methods for parabolic integro-differential equations, Math. Comp. 53 (1989), no. 187, 121–139. MR 969493, DOI 10.1090/S0025-5718-1989-0969493-9
R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, Calcolo 40 (2003), no. 3, 195–212. MR 2025602, DOI 10.1007/s10092-003-0073-2
Elizabeth G. Yanik and Graeme Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal. 12 (1988), no. 8, 785–809. MR 954953, DOI 10.1016/0362-546X(88)90039-9
Nai Ying Zhang, On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type, Math. Comp. 60 (1993), no. 201, 133–166. MR 1149295, DOI 10.1090/S0025-5718-1993-1149295-4
O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337–357. MR 875306, DOI 10.1002/nme.1620240206
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. MR 1161557, DOI 10.1002/nme.1620330702