基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

doi:10.12286/jssx.2012.2.125

计算数学 ›› 2012, Vol. 34 ›› Issue (2): 125-138.DOI: 10.12286/jssx.2012.2.125

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

王淑燕, 陈焕贞

山东师范大学数学科学学院, 济南 250014

收稿日期:2010-06-22 出版日期:2012-05-15 发布日期:2012-05-20
通讯作者: 陈焕贞
基金资助:
国家自然科学基金(10271068,10971254), 山东省自然科学基金(ZR2009AZ003),山东省优秀中青年科学家科研奖励基金(2008BS01008)资助项目.

PDF ( 1183 ) 引用被引次数 ( 3 )

王淑燕, 陈焕贞. 基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析[J]. 计算数学, 2012, 34(2): 125-138.

Wang Shuyan, Chen Huanzhen. AN IMMERSED FINITE ELEMENT METHOD BASED ON CROUZEIX-RAVIART ELEMENTS[J]. Mathematica Numerica Sinica, 2012, 34(2): 125-138.

AN IMMERSED FINITE ELEMENT METHOD BASED ON CROUZEIX-RAVIART ELEMENTS

Wang Shuyan, Chen Huanzhen

School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China

Received:2010-06-22 Online:2012-05-15 Published:2012-05-20

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.

关键词： 二阶椭圆界面问题, 浸入有限元, Crouzeix-Raviart 元, 最优误差估计

In this paper we present an immersed finite element method to solve numerically second order elliptic interface problems. The characteristics of the method is to prescribe a modified linear finite element space on each interface element in order to enforce the flux jump condition on the smooth interface, and a Crouzeix-Raviart non-conforming element on each non-interface element. Optimal-order error estimates are derived in the broken H¹?norm and L²?norm.

Key words: second order elliptic interface problems, the immersed finite element, Crouzeix- Raviart elements, optimal-order error estimates

MR(2010)主题分类:

65N15
65N30

分享此文:

[1] Babuska I. The finite element method for elliptic equations with discontinuous coefficients, Computing 5, 207-213, 1970.

[2] Bramble J H, King J T. A finite element method for interface problems in domains with smooth boundary and interfaces[J]. Adv. Comput. Math., 1996, 6: 109-138.

[3] Chen Z, Zou J. Finite element methods and their convergence for elliptic and parabolic interface problems[J]. Numer. Math., 1998, 79: 175-202.

[4] Efendiev Y R,Wu X H. Multiscale finite element for problems with highly oscillatory coefficients[J]. Numer. Math., 2002, 90: 459-486.

[5] Girault V, Raviart P A. Finite element methods for Navier-Stokes equations, Theory and Algorithms. Springer, Berlin, 1986.

[6] Gong Y, Li B, Li Z. Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions[J]. SIAM J. Numer. Anal., 2008, 46: 472-495.

[7] Hou T, Li Z, Osher S, Zhao H. A hybird method for moving interface problems with application to the Hele-Shaw flow[J]. J. Comput. Phys., 1997, 134: 236-252.

[8] Ladyzhenskaya O A, Rivkind V Ja, Ural’ceva N N, The classical solvability of diffraction problem, Trudy Mat. Inst. Steklov, vol. 92, pp. 116-146. Translated in Proceedings of the Steklov Institute of Math., no. 92, Boundary Value Problems of Mathematical Physics IV. American Mathematical Society, Providence, 1996.

[9] LeVeque R J, Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources[J]. SIAM J. Numer. Anal., 1994, 31: 1019-1044.

[10] LeVeque R J, Li Z. Immersed interface method for Stokes flow with elastic with boundaries or surface tension[J]. SIAM J. Sci. Comput., 1997, 18: 709-735.

[11] Lin T, Lin Y, Rogers R, Ryan M L. A rectangular immersed finite element space for interface problems[J]. Adv. Comput. Theory Pract., 2001: 7: 107-114.

[12] Li Z, Ito K. The Immersed Interface Method, Numerical solutions of PDEs involving interfaces and irregular domains. p. 176. SIAM, Philadelphia, 2006.

[13] Li Z, Lin Rogers R C. Error estimates of an immersed finite element method for interface problems, appearing in Methods Partial Differential Equations.

[14] Li Z, Lin T, Wu X. New Cartesian grid methods for interface problems using the finite element formulation[J]. Numer. Math., 2003: 96: 61-98.

[15] Li Z, Lin T Y, Rogers R C. An immersed finite element space and its approximation capability[J]. Numer. Methods Partial Differential Equations, 2004, 20: 338-367.

[16] Crouzeix M, Raviart P A. Conforming and nonconforming finite element methods for solving the stationary Stokes equations[J]. RAIRO Anal. Numer., 1973, 3: 33-75.

[17] Ciarlet P G. The finite element method for elliptic problems, Amsterdam, North-Holland, 1978.

[18] Roitberg J A, Seftel Z G. A theorem on homeomorphisms for elliptic systems and its applications[ J]. Math. USSR-Sb., 1969, 7: 439-465.

[19] Chou So-Hsiang, Kwak D Y, Wee K T. Optimal convergence analysis of an immersed interface finite element method. Springer Science and Business Media, 2009.

[20] 王烈衡, 许学军. 有限元方法的数学基础[M]. 北京: 科学出版社, 2004.

[1]	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.
[2]	石东洋, 张斐然. Sine-Gordon方程的一类低阶非协调有限元分析[J]. 计算数学, 2011, 33(3): 289-297.
[3]	苏剑, 李开泰. 混合边界条件下的三维定常旋转Navier-Stokes方程的原始变量有限元计算[J]. 计算数学, 2008, 30(3): 235-246.
[4]	李焕荣,罗振东,李潜,. 二维粘弹性问题的广义差分法及其数值模拟[J]. 计算数学, 2007, 29(3): 251-262.
[5]	李焕荣,罗振东,谢正辉,朱江,. 非饱和土壤水流问题的广义差分法及其数值模拟[J]. 计算数学, 2006, 28(3): 321-336.
[6]	石东洋,毛士鹏,陈绍春. 问题变分不等式的一类各向异性Crouzeix-Raviart型有限元逼近[J]. 计算数学, 2005, 27(1): 45-54.
[7]	江成顺,崔霞. 一类非线性反应扩散方程组的有限元分析[J]. 计算数学, 2000, 22(1): 103-112.

阅读次数

全文

摘要

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

AN IMMERSED FINITE ELEMENT METHOD BASED ON CROUZEIX-RAVIART ELEMENTS

扫码分享