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抛物型界面问题的变网格有限元方法

关宏波, 洪亚鹏   

  1. 郑州轻工业大学 数学与信息科学学院, 郑州 450002
  • 收稿日期:2018-07-04 出版日期:2020-05-15 发布日期:2020-05-15
  • 基金资助:

    国家自然科学基金(11501527),郑州轻工业大学青年骨干教师基金(2016XGGJS008)、博士基金(2015BSJJ070)及研究生科技创新项目(2018018)资助.

关宏波, 洪亚鹏. 抛物型界面问题的变网格有限元方法[J]. 计算数学, 2020, 42(2): 196-206.

Guan Hongbo, Hong Yapeng. FINITE ELEMENT METHODS WITH MOVING GRIDS FOR PARABOLIC INTERFACE PROBLEMS[J]. Mathematica Numerica Sinica, 2020, 42(2): 196-206.

FINITE ELEMENT METHODS WITH MOVING GRIDS FOR PARABOLIC INTERFACE PROBLEMS

Guan Hongbo, Hong Yapeng   

  1. College of Mathematics and Information Science, Zhengzhou University of Light Industry Zhengzhou 450002, China
  • Received:2018-07-04 Online:2020-05-15 Published:2020-05-15
本文针对抛物型界面问题,提出了一种线性三角形变网格有限元方法.其主要思路是针对空间变量采用有限元离散,对时间变量采用差分离散,但是不同时刻的有限元剖分网格可以不同.在不引入Ritz投影这一传统分析工具的情况下,得到了最优误差估计结果,使得证明过程更加简洁.给出的数值算例验证了理论分析的正确性.
In this paper, the linear triangular finite element methods with moving grids are discussed for the parabolic interface problems. The general idea is applying finite element method in space and choosing difference method with respect to the time variable, respectively, but the grids can be different when the time varies. The optimal order error estimates are obtained without introducing the Ritz projection, which is thought to be a conventional analysis tool. Thus, the analysis procedure is made to be more concise. Numerical examples are provided to verify the theoretical analysis.

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[1] Li J, Renardy Y, Renardy M. Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method[J]. Physics of Fluids, 2000, 12(2):269-282.

[2] Hetzer G, Meir A. On an interface problem with a nonlinear jump condition, numerical approximation of solutions[J]. International Journal of Numerical Analysis and Modeling, 2007, 4(3-4):519-530.

[3] Babuška I. The finite element method for elliptic equations with discontinuous coefficients[J]. Computing, 1970, 5(3):207-213.

[4] He X, Lin T, Lin Y. Immersed finite element methods for elliptic interface problems with nonhomogeneous jump conditions[J]. International Journal of Numerical Analysis and Modeling, 2011, 8(2):284-301.

[5] He X, Lin T, Lin Y. The convergence of the bilinear and linear immersed finite element solutions to interface problems[J]. Numerical Methods for Partial Differential Equations, 2012, 28(1):312-330.

[6] Zhang Q, Ito K, Li Z, Zhang Z. Immersed finite elements for optimal control problems of elliptic PDEs with interfaces[J]. Journal of Computational Physics, 2015, 298(C):305-319.

[7] Han H. The numerical solutions of interface problems by infinite element method[J]. Numerische Mathematik, 1982, 39(1):39-50.

[8] Barrett J, Elliott C. Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces[J]. IMA Journal of Numerical Analysis, 1987, 7(3):283-300.

[9] Chen Z, Zou J. Finite element methods and their convergence for elliptic and parabolic interface problems[J]. Numerische Mathematik, 1998, 79(2):175-202.

[10] Guan H, Shi D. P1-nonconforming triangular FEM for elliptic and parabolic interface problems[J]. Applied Mathematics and Mechanics, 2015, 36(9) 1197-1212.

[11] Sinha R, Deka B. A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems[J]. Calcolo, 2006, 43(4):253-278.

[12] Sinha R, Deka B. Finite element methods for semilinear elliptic and parabolic interface problems[J]. Applied Numerical Mathematics, 2009, 59(8):1870-1883.

[13] Huang J, Zou J. Some New a priori estimates for second-order elliptic and parabolic interface problems[J]. Journal of Differential Equations, 2002, 184(2):570-586.

[14] Li J, Melenk J, Wohlmuth B, Zou J. Optimal a priori estimates for higher order finite elements for elliptic interface problems[J]. Applied Numerical Mathematics, 2010, 60(1-2):19-37.

[15] 梁国平.变网格的有限元法[J].计算数学, 1985, 11(4):377-384

[16] 袁益让. 一类退化非线性抛物型方程组的变网格有限元方法[J]. 计算数学, 1986, 12(2):121-136

[17] Shi D, Zhang Y. A nonconforming anisotropic finite element approximation with moving grids for stokes problem[J]. Journal of Computational Mathematics, 2007, 24(5):561-578.

[18] Shi D, Guan H. A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes[J]. Hokkaido Mathematical Journal, 2007, 36(4):687-709.

[19] Brenner S, Scott L. The Mathematical Theory of Finite Element Methods[M]. Springer-Verlag, Berlin, 1994.

[20] Attanayake C, Senaratne D. Convergence of an immersed finite element method for semilinear parabolic interface problems[J]. Applied Mathematical Sciences, 2011, 5(3):135-147.

[21] Thomas A, Serge N, Joachim S. Crouzeix-Raviart type finite elements on anisotropic meshes[J]. Numerische Mathematik, 2001, 89(2):193-223.

[22] Kim S, Yim J, Sheen D. Stable cheapest nonconforming finite elements for the Stokes equations[J]. Journal of Computational and Applied Mathematics, 2016, 299:2-14.
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