Interpolated Coefficients Finite Element Method for Nonlinear Two-Point Boundary Value Problems
DOI: 10.23977/jpms.2022.020102 | Downloads: 8 | Views: 571
Lizheng Cheng 1, Hongping Li 1
1 Institute of Primary Education, Changsha Normal University, Changsha, 410081, China
Corresponding AuthorHongping Li
In this paper, we consider nonlinear two-point boundary value problem using the Interpolated Coefficients Finite Element Method (ICFEM). We use the slice k-degree polynomial interpolation for nonlinear term and use Newton's method to solve the nonlinear equation system. We find the error convergence order of the ICFEM has some obvious characteristics. When k is an odd number, the error order is the normal finite element convergence order. And when k is an even number, the error convergence order has super convergence. The numerical results show the error convergence order of the 3rd ICFEM at the nodes is basically the normal finite element convergence order k + 1, and k = 2,4, the error convergence order of the 4th ICFEM at the node is generally higher than k + 1, almost reaching the 2k order super.
KEYWORDSFinite element method, interpolated coefficients, nonlinear, convergence order
CITE THIS PAPER
Lizheng Cheng, Hongping Li, Interpolated Coefficients Finite Element Method for Nonlinear Two-Point Boundary Value Problems. Journal of Probability and Mathematical Statistics (2022) Vol. 2: 9-15. DOI: http://dx.doi.org/10.23977/jpms.2022.020102.
 T. Linss, "Lay er-adapted meshes for convection-diffusion problems", Comput. Methods Appl. Mech. Engrg., vol. 192, pp. 1061-1105, 2003.
 Y. Yin, P. Zhu, B. Wang, "Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem", Numerical Mathematics: Theory, Methods and Applications, vol. 010, no. 1, pp. 44-64, 2017.
 W. K. Zahra, D. M. Van, "Discrete Spline Solution of Singularly Perturbed Problem with Two Small Parameters on a Shishkin-Type Mesh", Computational Mathematics and Modeling, vol. 029, no. 3, pp. 367-381, 2018.
 A. Das, S. Natesan, "Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Shishkin mesh", Journal of Applied Mathematics and Computing, vol. 059, pp. 207-225, 2019.
 M. Brdar, H. Zarin, "A singularly perturbed problem with two parameters on a Bakhvalov-type mesh", Journal of Computational and Applied Mathematics, vol. 292, pp. 307-319, 2016.
 D. Shakti, J. Mohapatra, "Layer-adapted Meshes for Parameterized Singular Perturbation Problem", Procedia Engineering, vol.127, pp. 539-544, 2015.
 Z. Cen, J. Chen, L. Xi, "A second-order hybrid finite difference scheme for a system of coupled singularly perturbed initial value problems", Journal of Computational & Applied Mathematics, vol. 234, no. 12, pp. 3445-3457, 2010.
 R. C. Eberhart, J. Kennedy, "A new optimizer using particle swarm theory", Proceedings of the Sixth International Sympo- sium on Micro Machine and Human Science. Nagoya: IEEE, pp. 39-43, 1995.
 Li H., Cheng L. Particle Swarm Optimization algorithm for Solving Singular Perturbed Problemsation Problems on Layer Adaptive Mesh. 2021.