TY - JOUR
T1 - The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations
AU - Liang, Hui
AU - Brunner, Hermann
N1 - Funding Information:
The work of the author Hui Liang was supported by the National Nature Science Foundation of China (No. 11771128, 11101130), Fundamental Research Project of Shenzhen (JCYJ20190806143201649), Project (HIT.NSRIF.2020056) Supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, and Research start-up fund Foundation in Harbin Institute of Technology (20190019). The work of the author Hermann Brunner was supported by the Hong Kong Research Grants Council GRF Grants HKBU 200113 and 12300014.
PY - 2020/7
Y1 - 2020/7
N2 - 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 - α 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.
AB - 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 - α 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.
KW - Collocation methods
KW - Convergence
KW - Endpoint
KW - Mesh points
KW - Uniform meshes
KW - Volterra integral equations
KW - Weakly singular kernels
UR - http://www.scopus.com/inward/record.url?scp=85086989190&partnerID=8YFLogxK
U2 - 10.1007/s10915-020-01266-1
DO - 10.1007/s10915-020-01266-1
M3 - Journal article
AN - SCOPUS:85086989190
SN - 0885-7474
VL - 84
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 1
M1 - 12
ER -