Open Access
Issue |
ESAIM: M2AN
Volume 57, Number 2, March-April 2023
|
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Page(s) | 671 - 691 | |
DOI | https://doi.org/10.1051/m2an/2022100 | |
Published online | 27 March 2023 |
- M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000). [CrossRef] [Google Scholar]
- I. Babuška and W.C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems. SIAM J. Numer. Anal. 18 (1981) 565–589. [CrossRef] [MathSciNet] [Google Scholar]
- H. Brunner, On the numerical solution of nonlinear Volterra integro-differential equations. BIT Numer. Math. 13 (1973) 381–390. [CrossRef] [Google Scholar]
- H. Brunner, Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations. Math. Comput. 42 (1984) 95–109. [CrossRef] [Google Scholar]
- H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004). [Google Scholar]
- H. Brunner and P.J. van der Houwen, The Numerical Solution of Volterra Equations. North-Holland, Amsterdam (1986). [Google Scholar]
- H. Brunner and D. Schötzau, hp-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J. Numer. Anal. 44 (2006) 22–245. [Google Scholar]
- H. Brunner, A. Pedas and G. Vainikko, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39 (2001) 957–982. [CrossRef] [MathSciNet] [Google Scholar]
- T.A. Burton, Volterra Integral and Differential Equations, 2nd edition. Vol. 202, Mathematics in Science and Engineering, Elsevier B.V. Amsterdam (2005). [Google Scholar]
- C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Scientific Computation. Springer, Berlin, Heidelberg (2006). [CrossRef] [Google Scholar]
- W.X. Cao, Z.M. Zhang and Q.S. Zou, Is 2k-conjecture valid for finite volume methods? SIAM J. Numer. Anal. 53 (2015) 942–962. [CrossRef] [MathSciNet] [Google Scholar]
- W.X. Cao, H.L. Liu and Z.M. Zhang, Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations. Numer. Methods Partial Differ. Equ. 33 (2017) 290–317. [CrossRef] [Google Scholar]
- W.X. Cao, L.L. Jia and Z.M. Zhang, A C1 Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete Contin. Dyn. Syst. Ser. B 26 (2021) 81–105. [MathSciNet] [Google Scholar]
- Q. Hu, Stieltjes derivatives and β-polynomial spline collocation for Volterra integrodifferential equations with singularities. SIAM J. Numer. Anal. 33 (1996) 208–220. [CrossRef] [MathSciNet] [Google Scholar]
- Q.M. Huang and H.H. Xie, Superconvergence of Galerkin solutions for Hammerstein equations. Int. J. Numer. Anal. Model. 6 (2009) 696–710. [MathSciNet] [Google Scholar]
- H. Kaneko and Y. Xu, Superconvergence of the iterated Galerkin methods for Hammerstein equations. SIAM J. Numer. Anal. 33 (1996) 1048–1064. [CrossRef] [MathSciNet] [Google Scholar]
- T. Lin, Y.P. Lin, M. Rao and S.H. Zhang, Petrov-Galerkin methods for linear Volterra integro-differential equations. SIAM J. Numer. Anal. 38 (2000) 937–963. [CrossRef] [MathSciNet] [Google Scholar]
- T. Lin, Y.P. Lin, P. Luo, M. Rao and S.H. Zhang, Petrov-Galerkin methods for nonlinear Volterra integro-differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. B 8 (2001) 405–426. [Google Scholar]
- P. Linz, Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia, PA (1985). [CrossRef] [Google Scholar]
- Ch Lubich, Runge-Kutta theory for Volterra integro-differential equations. Numer. Math. 40 (1982) 119–135. [CrossRef] [MathSciNet] [Google Scholar]
- K. Mustapha, A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels. Math. Comput. 82 (2013) 1987–2005. [CrossRef] [Google Scholar]
- K. Mustapha and J.K. Ryan, Post-processing discontinuous Galerkin solutions to Volterra integro-differential equations: analysis and simulations. J. Comput. Appl. Math. 253 (2013) 89–103. [CrossRef] [MathSciNet] [Google Scholar]
- I.P. Natanson, Theory of Functions of a Real Variable, Translated from the Russian by Leo F. Boron with the collaboration of Edwin Hewitt. Frederick Ungar Publishing Co., New York (1955). [Google Scholar]
- T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J. Numer. Anal. 13 (1993) 93–99. [CrossRef] [MathSciNet] [Google Scholar]
- Z.Q. Wang, Y.L. Guo and L.J. Yi, An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math. Comput. 86 (2017) 2285–2324. [CrossRef] [Google Scholar]
- A.M. Wazwaz, Linear and Nonlinear Integral Equations, Methods and Applications. Higher Education Press, Beijing, Springer, Heidelberg (2011). [CrossRef] [Google Scholar]
- J. Wen, C.M. Huang and M. Li, Stability analysis of Runge-Kutta methods for Volterra integro-differential equations. Appl. Numer. Math. 146 (2019) 73–88. [CrossRef] [MathSciNet] [Google Scholar]
- Y.X. Wei and Y.P. Chen, Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations. Numer. Math. Theory Methods Appl. 4 (2011) 419–438. [CrossRef] [MathSciNet] [Google Scholar]
- L.J. Yi, An h-p version of the continuous Petrov-Galerkin finite element method for nonlinear Volterra integro-differential equations. J. Sci. Comput. 65 (2015) 715–734. [CrossRef] [MathSciNet] [Google Scholar]
- L.J. Yi and B.Q. Guo, An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations. Sci. China Math. 57 (2014) 2285–2300. [CrossRef] [MathSciNet] [Google Scholar]
- L.J. Yi and B.Q. Guo, An h-p version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53 (2015) 2677–2704. [CrossRef] [MathSciNet] [Google Scholar]
- W. Yuan and T. Tang, The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation. Math. Comput. 54 (1990) 155–168. [CrossRef] [Google Scholar]
- S.H. Zhang, T. Lin, Y.P. Lin and M. Rao, Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations. Appl. Math. 45 (2000) 241–263. [CrossRef] [MathSciNet] [Google Scholar]
- S.H. Zhang, T. Lin, Y.P. Lin and M. Rao, Extrapolation and a-posteriori error estimators of Petrov-Galerkin methods for non-linear Volterra integro-differential equations. J. Comput. Math. 19 (2001) 407–422. [MathSciNet] [Google Scholar]
- M.Z. Zhang, X.Y. Mao and L.J. Yi, Exponential convergence of the hp-version of the composite Gauss-Legendre quadrature for integrals with endpoint singularities. Appl. Numer. Math. 170 (2021) 340–352. [CrossRef] [MathSciNet] [Google Scholar]
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