The fine error estimation of collocation methods on uniform meshes for weakly singular Volterra integral equations. (English) Zbl 1456.65184
Summary: It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree \(m\) to solve it numerically, due to the weak singularity of the solution at the initial time \(t=0\), only \(1-\alpha\) global convergence order can be obtained on uniform meshes, comparing with \(m\) global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as \(n\) increasing. In particular, 1 order can be recovered for \(m=1\) at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 45E99 | Singular integral equations |
Keywords:
Volterra integral equations; weakly singular kernels; collocation methods; uniform meshes; convergence analysisReferences:
| [1] | Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1059.65122 |
| [2] | Cao, YZ; Herdman, T.; Xu, YS, A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM J. Numer. Anal., 41, 364-381 (2003) · Zbl 1042.65106 · doi:10.1137/S0036142901385593 |
| [3] | Dixon, J.; McKee, S., Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech., 66, 535-544 (1986) · Zbl 0627.65136 · doi:10.1002/zamm.19860661107 |
| [4] | Liang, H.; Brunner, H., The convergence of collocation solutions in continuous piecewise polynomial spaces for weakly singular Volterra integral equations, SIAM J. Numer. Anal., 57, 4, 1875-1896 (2019) · Zbl 1420.65139 · doi:10.1137/19M1245062 |
| [5] | Liu, X.; Mckee, S.; Yuan, JY; Yuan, YX, Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices, Linear Algebra Appl., 435, 1157-1170 (2011) · Zbl 1230.15002 · doi:10.1016/j.laa.2011.02.044 |
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.
