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# Article

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Keywords:
Volterra integro-differential equations; Petrov-Galerkin methods; asymptotic expansions; defect correction; a posteriori error estimators
Summary:
We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to forma posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.
References:
[1] H. Brunner: A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations. J. Comput. Appl. Math. 8 (1982), 213–229. MR 0682889 | Zbl 0485.65087
[2] H. Brunner: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20 (1983), 1106–1119. MR 0723827 | Zbl 0533.65087
[3] H. Brunner: Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations. Math. Comp. 42, 165 (1984), 95–109. MR 0725986 | Zbl 0543.65092
[4] H. Brunner: Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels. IMA J. Numer. Anal. 6 (1986), 221–239. MR 0967664 | Zbl 0634.65142
[5] H. Brunner: The numerical treatment of Volterra integro-differential equations with unbounded delay. J. Comput. Appl. Math. 28 (1989), 5–23. MR 1038829 | Zbl 0687.65131
[6] H. Brunner, P. J. van der Houwen: The Numerical Solution of Volterra Equations. CWI Monographs, Vol. 3, North-Holland, Amsterdam, 1986. MR 0871871
[7] H. Brunner, J.-P. Kauthen and A. Ostermann: Runge-Kutta time discretizations of parabolic Volterra integro-differential equations. J. Integral Equations Appl. 7 (1995), 1–16. MR 1339686
[8] H. Brunner, Y. Lin and S. Zhang: Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods. J. Integral Equations Appl. 10 (1998), 375–396. MR 1669667
[9] H. Brunner, A. Makroglou and R. K. Miller: Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl. Numer. Math. 23 (1997), 381–402. MR 1453423
[10] J. Cannon, Y. Lin: A priori $L^2$ error estimates for finite element methods for nonlinear diffusion equations with memory. SIAM J. Numer. Anal. 27 (1990), 595–607. MR 1041253
[11] Q. Hu: Stieltjes derivatives and $\beta$-polynomial spline collocation for Volterra integro-differential equations with singularities. SIAM J. Numer. Anal. 33 (1996), 208–220. MR 1377251
[12] J. P. Kauthen: The method of lines for parabolic partial integro-differential equations. J. Integral Equations Appl. 4 (1992), 69–81. MR 1160088 | Zbl 0763.65101
[13] M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Essex, 1990. MR 1066462
[14] M. Křížek, P. Neittaanmäki: Bibliography on Superconvergence. In: Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates (edited by M. Křížek, P. Neittaanmäki and R. Stenberg), Lecture Notes in Pure and Applied Mathematics, vol. 196. Marcel Dekker, New York, 1998, pp. 315–348. MR 1602809
[15] Q. Lin, N. Yan: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers, 1996.
[16] Q. Lin, S. Zhang: An immediate analysis for global superconvergence for integro-differential equations. Appl. Math. 42 (1997), 1–21. MR 1426677
[17] Q. Lin, S. Zhang and N. Yan: Methods for improving approximate accuracy for hyperbolic integro-differential equations. Systems Sci. Math. Sci. 10 (1997), 282–288. MR 1469188
[18] Q. Lin, S. Zhang and N. Yan: An acceleration method for integral equations by using interpolation post-processing. Adv. in Comput. Math. 9 (1998), 117–128. MR 1662762
[19] T. Lin, Y. Lin, M. Rao and S. Zhang: Petrov-Galerkin methods for nonlinear Volterra integro-differential equations. Submitted to J. Math. Anal. Appl.
[20] R. K. Miller: An integro-differential equation for rigid heat conductions with memory. J. Math. Anal. Appl. 66 (1978), 313–332. MR 0515894
[21] J. Prüss: Evolutionary Integral Equations and Applications. Birkhauser Verlag, Basel, 1993. MR 1238939
[22] T. Tang: Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations. Numer. Math. 61 (1992), 373–382. MR 1151776 | Zbl 0741.65110
[23] T. Tang: A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 13 (1993), 93–99. MR 1199031 | Zbl 0765.65126

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