Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind. (English) Zbl 1476.65340
Summary: In this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution \(u_h\), two different interpolation postprocessing approximations of higher accuracy: \(I_{2h}^{2m-1}u_h\) based on the collocation points and \(I_{2h}^m u_h\) based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
Keywords:
Volterra integral equations; superconvergence; supercloseness; interpolation postprocessing; weakly singular kernels; collocation; hybrid collocationReferences:
| [1] | Atkinson, K., The numerical solution of integral equations of the second kind, 100-156 (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0899.65077 |
| [2] | Brunner, H., Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J Numer Anal, 20, 1106-1119 (1983) · Zbl 0533.65087 |
| [3] | Brunner, H., Iterated collocation methods and their discretizations for Volterra integral equations, SIAM J Numer Anal, 21, 1132-1145 (1984) · Zbl 0575.65134 |
| [4] | Brunner, H., The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math Comp, 45, 417-437 (1985) · Zbl 0584.65093 |
| [5] | Brunner, H.; Yan, N., On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations, J Comput Math, 10, 348-357 (1992) · Zbl 0758.65083 |
| [6] | Brunner, H., Collocation methods for Volterra integral and related functional differential equations, 53-418 (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1059.65122 |
| [7] | Cao, Y.; Herdman, T.; Xu, Y., A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J Numer Anal, 41, 364-381 (2003) · Zbl 1042.65106 |
| [8] | Diogo, T., Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, J Comput Appl Math, 229, 363-372 (2009) · Zbl 1168.65073 |
| [9] | Diogo, T.; McKee, S.; Tang, T., Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proc R Soc Edinb, 124, 199-210 (1994) · Zbl 0807.65141 |
| [10] | Graham, I.; Joe, S.; Sloan, IH, Iterated Galerkin versus iterated collocation for integral equations of the second kind, IMA J Numer Anal, 5, 355-369 (1985) · Zbl 0586.65091 |
| [11] | Hu, Q., Stieltjes derivatives and \(\beta \)-polynomial spline collocation for Volterra integro-differential equations with singularities, SIAM, J Numer Anal, 33, 208-220 (1996) · Zbl 0851.65098 |
| [12] | Hu, Q., Superconvergence of numerical solutions to Volterra integral equations with singularities, SIAM J Numer Anal, 34, 1698-1707 (1997) · Zbl 0892.65083 |
| [13] | Huang, Q.; Zhang, S., Superconvergence of interpolated collocation solutions for Hammerstein equations, Numer Methods Partial Differ Equ, 26, 290-304 (2010) · Zbl 1191.65178 |
| [14] | Huang, Q.; Xie, H., Superconvergence of the interpolated Galerkin solutions for Hammerstein equations, Int J Numer Anal Model, 6, 696-710 (2009) · Zbl 1499.65746 |
| [15] | Huang, M.; Xu, Y., Superconvergence of the iterated hybrid collocation method for weakly singular Volterra integral equations, J Integral Equ Appl, 18, 83-116 (2006) · Zbl 1141.65091 |
| [16] | Kaneko, H.; Xu, Y., Gaussian-type quadratures for weakly singular integrals and their applications to the Fredholm integral equation of the second kind, Math Comput, 62, 739-753 (1994) · Zbl 0799.65023 |
| [17] | Lin, Q.; Lin, J., Finite element methods: accuracy and improvement (2006), Beijing: Science Press, Beijing |
| [18] | Lin Q, Yan N, (1996) The construction and analysis of high efficiency finite element methods. Hebei University Publishers (in Chinese) |
| [19] | Lin, Q.; Zhang, S.; Yan, N., An acceleration method for integral equations by using interpolation post-processing, Adv Comp Math, 9, 117-129 (1998) · Zbl 0920.65087 |
| [20] | Naga, A.; Zhang, Z., A posteriori error estimates based on the polynomial preserving recovery, SIAM J Numer Anal, 42, 1780-1800 (2004) · Zbl 1078.65098 |
| [21] | Pedas, A.; Vainikko, G., Numerical solution of weakly singular Volterra integral equations with change of variables, Proc Eston Acad Sci Phys Math, 53, 96-106 (2004) · Zbl 1065.65149 |
| [22] | Pedas, A.; Vainikko, G., Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations, Computing, 73, 271-293 (2004) · Zbl 1063.65147 |
| [23] | Rebelo, M.; Diogo, T., A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel, J Comput Appl Math, 234, 2859-2869 (2010) · Zbl 1196.65202 |
| [24] | Sloan, IH, Improvement by iteration for compact operator equations, Math Comput, 30, 758-764 (1976) · Zbl 0343.45010 |
| [25] | Sloan, IH; Golberg, M., Superconvergence, Numerical solution of integral equations, 35-70 (1990), New York: Plenum, New York · Zbl 0759.65091 |
| [26] | Tang, T., Superconvergence of numerical solutions to weakly singular Volterra Integral differential equations, Numer Math, 61, 373-382 (1992) · Zbl 0741.65110 |
| [27] | Zhang, S.; Lin, Y.; Rao, M., Numerical solutions for second-kind Volterra integral equations by Galerkin methods, Appl Math, 45, 19-39 (2000) · Zbl 1058.65148 |
| [28] | Zhang, Z.; Naga, A., A new finite element gradient recovery method: superconvergence property, SIAM J Sci Comput, 26, 1192-1213 (2005) · Zbl 1078.65110 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
