## Numerical solutions for second-kind Volterra integral equations by Galerkin methods.(English)Zbl 1058.65148

The authors consider a nonlinear Volterra integral equation of the second kind in the interval [0,1]. They prove that a simple interpolated finite element Galerkin scheme produces a global superconvergence in the $$L^{\infty}$$-norm. A local superconvergence of iterated finite element solutions is investigated, as well. As a by-product, an a posteriori error estimator is obtained.

### MSC:

 65R20 Numerical methods for integral equations 45L05 Theoretical approximation of solutions to integral equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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### References:

 [1] K. Atkinson, J. Flores: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13 (1993), 195-213. · Zbl 0771.65090 [2] H. Brunner: Iterated collocation methods and their discretization for Volterra integral equations. SIAM J. Numer. Anal. 21 (1984), 1132-1145. · Zbl 0575.65134 [3] H. Brunner: The approximate solution of Volterra equations with nonsmooth solutions. Utilitas Math. 27 (1985), 57-95. · 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. · 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. · Zbl 0611.65092 [6] H. Brunner, Q. Lin, N. Yan: The iterative correction method for Volterra integral equations. BIT 36:2 (1996), 221-228. · Zbl 0854.65121 [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. · Zbl 0944.65140 [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. · Zbl 0941.65136 [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. · Zbl 0857.65145 [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. · Zbl 0851.65098 [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. · Zbl 0708.65106 [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. · Zbl 0724.65128 [13] Q. Lin, S. Zhang: An immediate analysis for global superconvergence for integrodifferential equations. Appl. Math. 1 (1997), 1-21. · Zbl 0902.65090 [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. · Zbl 0899.65081 [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. · Zbl 0920.65087 [16] I. H. Sloan: Superconvergence. Numerical Solution of Integral Equations, M. A. Golberg (ed.), Plenum Press, New York, 1990, pp. 35-70. · Zbl 0759.65091
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