一类椭圆型界面问题的数值算法

Abstract

Abstract:

In this paper, a numerical algorithm based on the theory of reproducing kernel space is proposed for the one-dimensional ellipse interface problem. This method integrates the reproducing kernel function of ${W}^{1}_{2}$ space to obtain a set of new basis in the cubic spline space, on which a broken cubic spline space is established. Then we use the least squares method to get the approximate solution of this kind of interface problem. We discussed the order of convergence under the $H_2$ norm, $H_1$ norm and $L_2$ norm. Finally, our theory is verified by several numerical example.

Key words: Interface problem, Reproducing kernel, The cubic spline space, The least squares method, Convergence analysis

CLC Number:

O241.8

Yu Yikang,Niu Jing. A Kind of Numerical Algorithm for Elliptic Interface Problem[J].Acta mathematica scientia,Series A, 2023, 43(3): 883-895.

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References 33

[1]	Greengard L, Moura M. On the numerical evaluation of electrostatic fields in composite materials. Acta Numer, 1994, 3: 379-410 doi: 10.1017/S0962492900002464
[2]	Peskin C. Numerical analysis of blood flow in the heart. J Comput Phys, 1977, 25: 220-252
[3]	Al-Smadi M, Arqub O A. Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates. Appl Math Comput, 2019, 342: 280-294 doi: 10.1016/j.amc.2018.09.020
[4]	Al-Smadi M, Arqub O A, Shawagfeh N, Momani S. Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method. Appl Math Comput, 2016, 291: 137-148 doi: 10.1016/j.amc.2016.06.002
[5]	Albright J, Epshteyn Y, Medvinsky M, Xia Q. High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces. Appl Numer Math, 2017, 111: 64-91 doi: 10.1016/j.apnum.2016.08.017
[6]	Babuska I, Caloz G, Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J Numer Anal, 1994, 31: 945-981 doi: 10.1137/0731051
[7]	Epshteyn Y, Phippen S. High-order difference potentials methods for 1D elliptic type models. Appl Numer Math, 2015, 93: 69-86 doi: 10.1016/j.apnum.2014.02.005
[8]	Hellrung J, Wang L, Sifakis E, Teran J. A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J Comput Phys, 2012, 231: 2015-2048 doi: 10.1016/j.jcp.2011.11.023
[9]	Lee C, Park J. Preconditioning for finite element methods with strain smoothing. Computers & Mathematics with Applications, 2023, 130: 41-57 doi: 10.1016/j.camwa.2022.11.018
[10]	Kalachikova U, Vasilyeva M, Harris I, et al. Generalized Multiscale Finite Element Method for scattering problem in heterogeneous media. J Comput Appl Math, 2023, 424: 114977 doi: 10.1016/j.cam.2022.114977
[11]	Guzman J, Sanchez M, Sarkis M. On the accuracy of finite element approximations to a class of interface problems. Math Comput, 2016, 85: 2071-2098 doi: 10.1090/mcom/2016-85-301
[12]	Harari I, Dolbow J. Analysis of an efficient finite element method for embedded interface problems. Comput Math, 2010, 46: 205-211
[13]	Dong B Y, Feng X F, Lin Z L. An FE-FD method for anisotropic elliptic interface problems. SIAM J Sci Comput, 2020, 42: B1041-B1066 doi: 10.1137/19M1291030
[14]	Pan K J, Wu X X, Xu Y F, Yuan G W. An exact-interface-fitted mesh generator and linearity-preserving finite volume scheme for anisotropic elliptic interface problems. J Comput Phys, 2022, 463: 111293 doi: 10.1016/j.jcp.2022.111293
[15]	Cao W, Zhang X, Zhang Z. Superconvergence of immersed finite element methods for interface problems. Adv Comput Math, 2017, 43: 795-821 doi: 10.1007/s10444-016-9507-7
[16]	He X, Lin T, Lin Y. Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int J Numer Anal Model, 2011, 8: 284-301
[17]	Cao W, Zhang X, Zhang Z, Zou Q. Superconvergence of immersed finite volume methods for one-dimensional interface problems. J Sci Comput, 2017, 73: 543-565 doi: 10.1007/s10915-017-0532-6
[18]	Liu K, Zou Q. Analysis of a special immersed finite volume method for elliptic interface problems. Int J Numer Anal Model, 2019, 16: 964-984
[19]	Liu W, Jun S, Zhang Y. Reproducing kernel particle method. Int J Numer Methods Fluids, 1995, 20: 1081-1106 doi: 10.1002/(ISSN)1097-0363
[20]	Yang J, Guan P, Fan C. Weighted reproducing kernel collocation method and error analysis for inverse Cauchy problems. Int J Appl Mech, 2016, 8: 1650030 doi: 10.1142/S1758825116500307
[21]	Yang J, Hsin W. Weighted reproducing kernel collocation method based on error analysis for solving inverse elasticity problems. Acta Mech, 2019, 230: 3477-3497 doi: 10.1007/s00707-019-02473-0
[22]	Chen J, Wang H, Hu H. Reproducing kernel enhanced local radial basis collocation method. Int J Numer Methods Eng, 2008, 80: 600-627
[23]	Niu J, Xu M, Yao G M. An efficient reproducing kernel method for solving the allen-cahn equation. Appl Math Lett, 2019, 89: 78-84 doi: 10.1016/j.aml.2018.09.013
[24]	Niu J, Sun L X, Xu M Q, Hou J J. A reproducing kernel method for solving heat conduction equations with delay. Appl Math Lett, 2020, 100: 106036 doi: 10.1016/j.aml.2019.106036
[25]	Niu J, Jia Y T, Sun J D. A new piecewise reproducing kernel function algorithm for solving nonlinear hamiltonian systems. Appl Math Lett, 2023, 136: 108451 doi: 10.1016/j.aml.2022.108451
[26]	Sun L X, Niu J, Hou J J. A high order convergence collocation method based on the reproducing kernel for general interface problems. Appl Math Lett, 2021, 112: 106718 doi: 10.1016/j.aml.2020.106718
[27]	Yu Y, Niu J, Zhang J, Ning S Y. A reproducing kernel method for nonlinear C-q-fractional IVPs. Appl Math Lett, 2022, 125: 107751 doi: 10.1016/j.aml.2021.107751
[28]	Yu Y K, Niu J, Yang X M, Cui M G. A broken reproducing kernel method for the multiple interface problems, Comput Appl Math, 2022, 41: 260
[29]	吴勃英, 林迎珍. 应用型再生核空间. 北京: 科学出版社, 2012
	Wu B, Lin Y Z. Applied Regenerative Nuclear Space. Beijing: Science Press, 2012
[30]	杨学敏, 牛晶, 姚春华. 椭圆型界面问题的破裂再生核方法. 计算数学, 2022, 44(2): 217-232 doi: 10.12286/jssx.j2021-0791
	Yang X M, Niu J, Yao C H. The method of fracture regeneration core for elliptical interface problem. Computational Mathematics, 2022, 44(2): 217-232 doi: 10.12286/jssx.j2021-0791
[31]	Hall C A, Meyer W W. Optimal error bounds for cubic spline interpolation. J Approx Theory, 1976, 16: 105-122 doi: 10.1016/0021-9045(76)90040-X
[32]	Xu M, Zhao Z, Niu J, Lin Y. A simplified reproducing kernel method for 1-D elliptic type interface problems. J Comput Appl Math, 2019, 351: 29-40 doi: 10.1016/j.cam.2018.10.027
[33]	林群, 谢和虎. 有限元Aubin-Nitsche技巧新认识及其应用. 数学的实践与认识, 2011, 41(17): 247-258
	Lin Q, Xie H. An observation on Aubin-Nitsche lemma and its applications. Mathematics in Practice and Theory, 2011, 41(17): 247-258

A Kind of Numerical Algorithm for Elliptic Interface Problem

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