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Volterra integral equations; Galerkin methods; convergence and superconvergence; interpolation post-processing; iterative correction; a posteriori error estimators
In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the $O(h^{2r})$-convergence rate in the piecewise-polynomial space of degree not exceeding $(r-1)$. As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.
[1] K. Atkinson, J. Flores: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13 (1993), 195–213. DOI 10.1093/imanum/13.2.195 | MR 1210822
[2] H. Brunner: Iterated collocation methods and their discretization for Volterra integral equations. SIAM J. Numer. Anal. 21 (1984), 1132–1145. DOI 10.1137/0721070 | MR 0765511
[3] H. Brunner: The approximate solution of Volterra equations with nonsmooth solutions. Utilitas Math. 27 (1985), 57–95. MR 0804372 | Zbl 0563.65077
[4] 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. DOI 10.1016/0771-050X(82)90044-4 | MR 0682889 | Zbl 0485.65087
[5] H. Brunner, P. J. Van der Houwen: The Numerical Solution of Volterra Equations. CWI Monographs, Vol. 3. North-Holland, Amsterdam, 1986. MR 0871871
[6] H. Brunner, Q. Lin, N. Yan: The iterative correction method for Volterra integral equations. BIT 36:2 (1996), 221–228. DOI 10.1007/BF01731980 | MR 1432245
[7] H. Brunner, Y. Lin, S. Zhang: Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods. J. Integ. Eqs. Appl 10, 4 (1998), 375–396. MR 1669667
[8] H. Brunner, A. Pedas, G. Vainikko: The piecewise polynomial collocation methods for nonlinear weakly singular Volterra equations. Research Reports A 392, Institute of Mathematics, Helsinki University of Technology, 1997.
[9] H. Brunner, N. Yan: On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations. J. Comput. Appl. Math. 67 (1996), 187–189. DOI 10.1016/0377-0427(96)00012-X | MR 1388148
[10] Q. Hu: Stieltjes derivatives and $\beta $-polynomial spline collocation for Volterra integro-differential equations with singularities. SIAM J. Numer. Anal. 33, 1 (1996), 208–220. DOI 10.1137/0733012 | MR 1377251
[11] 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
[12] Q. Lin, I.H. Sloan, R. Xie: Extrapolation of the iterated-collocation method for integral equations of the second kind. SIAM J. Numer. Anal. 27, 6 (1990), 1535–1541. DOI 10.1137/0727090 | MR 1080337
[13] Q. Lin, S. Zhang: An immediate analysis for global superconvergence for integrodifferential equations. Appl. Math. 1 (1997), 1–21. DOI 10.1023/A:1022264125558 | MR 1426677
[14] Q. Lin, S. Zhang, N. Yan: Methods for improving approximate accuracy for hyperbolic integrodifferential equations. Systems Sci. Math. Sci., 10, 3 (1997), 282–288. MR 1469188
[15] Q. Lin, S. Zhang, N. Yan: An acceleration method for integral equations by using interpolation post-processing. Advances in Comput. Math. 9 (1998), 117–129. DOI 10.1023/A:1018925103993 | MR 1662762
[16] I. H. Sloan: Superconvergence. Numerical Solution of Integral Equations, M. A. Golberg (ed.), Plenum Press, New York, 1990, pp. 35–70. MR 1067150 | Zbl 0759.65091
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