混合有限元的区域分解算法
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摘要
区域分解算法是求解偏微分方程最有效的数值计算方法之一,通常它主要应用于标准的有限元方法。本文主要考虑将其应用于非标准的有限元,即混合有限元方法的情形。本文共分五章,第1章首先介绍了有限元及将其应用于偏微分方程求解的步骤,然后引入混合有限元和区域分解算法及本文所要解决的主要问题。第2章介绍混合有限元的基础理论,我们介绍了混合有限元的存在唯一性理论,然后给出了一些方程的混合有限元格式及它们的解的存在唯一性的结论。第3章介绍区域分解算法,我们介绍了重叠型和非重叠型区域分解算法,对于重叠型区域分解算法,我们介绍了基于Lions框架的Schwarz交替法,对于非重叠型区域分解算法,我们介绍了基于Steklov-Poincare算子的D-N交替法。第4章介绍混合有限元的区域分解算法,我们引入一种基于Schwarz交替法的区域分解算法,给出了计算格式及相应的有限元离散,并且在分别基于Lions框架和极值原理的基础上,证明了它在不同范数下的收敛性。第5章,我们作了简要的总结及阐述了下一步的工作。本文表明区域分解算法也适用于混合有限元。
Domain decomposition method is one of the most effective numeric methods on solving partial differential equations. It is generally used with normative finite element method. In this paper, we mainly consider applying it to non-normative finite element method which is called mixed finite element method. The paper contains five chapters. In chapter 1, we first introduce finite element and the steps how to apply it to solve partial differential equation, then bring in mixed finite element and domain decomposition method, finally, we rise the main problem what we want to solve in the paper. In chapter 2, we introduce the basic theories about mixed finite element. We introduce the existence and uniqueness theorem of mixed finite element, and then show mixed finite element formula about some equations and the conclusion whether the solution of the formula is existence and uniqueness. In chapter 3, we introduce domain decomposition method. We introduce superposition and non-superposition domain decomposition method apart. About superposition domain decomposition method, based on Lions framework, we introduce Schwarz alternating method, about non-superposition domain decomposition method, based on Steklov-Poincare operator, we introduce D-N alternating method. In chapter 4, we introduce domain decomposition method of mixed finite element method. We introduce a method which based on Schwarz alternating method, we show the computing formula and relative discrete finite element, and based on Lions framework and extremum principle respectively, we prove its convergence in different norms. In chapter 5, we make a brief summarize and what we will do in next phase. In this paper, it is showed that domain decomposition method can also be used with mixed finite element.
Domain decomposition method is one of the most effective numeric methods on solving partial differential equations. It is generally used with normative finite element method. In this paper, we mainly consider applying it to non-normative finite element method which is called mixed finite element method. The paper contains five chapters. In chapter 1, we first introduce finite element and the steps how to apply it to solve partial differential equation, then bring in mixed finite element and domain decomposition method, finally, we rise the main problem what we want to solve in the paper. In chapter 2, we introduce the basic theories about mixed finite element. We introduce the existence and uniqueness theorem of mixed finite element, and then show mixed finite element formula about some equations and the conclusion whether the solution of the formula is existence and uniqueness. In chapter 3, we introduce domain decomposition method. We introduce superposition and non-superposition domain decomposition method apart. About superposition domain decomposition method, based on Lions framework, we introduce Schwarz alternating method, about non-superposition domain decomposition method, based on Steklov-Poincare operator, we introduce D-N alternating method. In chapter 4, we introduce domain decomposition method of mixed finite element method. We introduce a method which based on Schwarz alternating method, we show the computing formula and relative discrete finite element, and based on Lions framework and extremum principle respectively, we prove its convergence in different norms. In chapter 5, we make a brief summarize and what we will do in next phase. In this paper, it is showed that domain decomposition method can also be used with mixed finite element.
引文
[1]有限元方法的数学基础[M].王烈衡,许学军.北京,科学出版社.2004
[2]有限元方法及其应用[M].李开泰,黄艾香,黄庆怀.北京,科学出版社.2006
[3]偏微分方程现代数值解法[M].马逸尘,梅立泉,王阿霞.北京,科学出版社.2006
[4]偏微分方程数值解法[M].李荣华.北京,高等教育出版社.2004
[5]混合有限元基础及其应用[M].罗振东.北京,科学出版社.2006
[6]区域分解算法[M].吕涛,石济民,林振宝.北京,科学出版社.1992
[7]重调和方程的混合有限元的区域分解算法[J].王寿成,姚海波.佳木斯大学学报,1(2009), 579-581
[8]二阶椭圆问题的混合有限元分析[J].罗振东.应用数学,1992,5卷4期,26-31.
[9]三维空间的Stokes问题的混合有限元法[J].罗振东.贵州师范大学学报,1998,9卷2期,70-74.
[10]有限元混合法理论基础及其应用[J].发展与应用.山东教育出版社,济南,1996
[11] Stokes问题的混合有限元分析[J].王烈衡.计算数学,1987,9(1),70-81.
[12] Analysis of some mixed finite element methods for the Stokes problem [J].Bernardi Cand Rangel C.Math.Comput, 1985, Vol.44 No.169, 71-79.
[13] Error-bounds for finite element method [J].Babuska.I.Numer.Math.1971, Vol.16,322-33
[14] On the existence, uniqueness and approximation of Saddle point problems arising from Lagrangian multipliers [J]. RAIRO Anal. Numer.,1972,8,129-151.
[15] Mixed and hybrid Finite Element Methods [M]. F.Brezzi, M.Fortin, Springer-rerlag, 1991.
[16] A mixed Finite Element Method for the Biharmnic Equation, Symposium on Mathematical Aspects of Finite Element in Partial Differential Equations [M]. Ciarlet P G and Rariart P. C de Boor, Ed., New York:Academic press, 1974,125-143.
[17] Finite Element Methods for Navier-Stokes Equations [M]. V.Girault and P.Raviart. Berlin:Springer,1986.
[18] Schwarz算法的Lions框架与异步并行算法的收敛性证明[J].吕涛.系统科学与数学,9(2),1989,128-132.
[19]椭圆型方程区域分解方法中某些问题研究[J].储德林,博士论文,1991.
[20]关于解椭圆型问题的两个子区域不重叠区域分解算法[J].顾金生,胡显承.计算数学,4(1664),432-447.
[21]基于非协调元的区域分解算法[J],黄建国.计算数学,1(1995),47-58.
[22] On the convergence of a space decomposition methods for nonlinear problems [J].周叔子,曾金平.数学进展.28(1999),541-542.
[23]非嵌套网格上的Morley元两水平加性Schwarz方法[J].石钟慈,谢正辉.计算数学,3(
[24] On Schwarz alternating method.Ⅰ. Lions,P. Proceedings of First Internations Symposium on Domain Decomposition Methods for Patial Differential Equations, SIAM,Philadelphia,PA,1988.
[25] On Schwarz alternating method.Ⅱ. Lions,P. Domain decomposition methods, SIAM, Philadelphia,PA,1989.
[26] Two synchronous parallel algorithms for partial differential equations [J]. Lu,T., Shih,T.M. and Liem,C.B. J.Comp.Math.,9(1991),74-85.
[27]一个双调和方程的Schwarz交替法[J].蒋美群,邓庆平.计算数学,1994,(1):93-101
[28]椭圆与抛物形方程引论[M].伍卓群.北京:科学出版社, 2005,1-50.
[29] Elliptic Partial Differential Equations of Second Order [M]. D.Glibarg and N.S.Trudinger, Springer verlag, 1977.
[30]有限元逼近的离散强极值原理与Schwarz算法的收敛性[J].沈树民,蒋美群,邓庆平.高等学校计算数学学报,4(1995),339-344.
[31] Analysis of Schwarz algorithms for mixed finite element methods. Ewing,R. and Wang,J. RAIRO,MAN, 1992 (26):739-756.
[32] Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems:algorithms and numerical results, Mathew,T. Numer,math., 1993 (65):445-492
[2]有限元方法及其应用[M].李开泰,黄艾香,黄庆怀.北京,科学出版社.2006
[3]偏微分方程现代数值解法[M].马逸尘,梅立泉,王阿霞.北京,科学出版社.2006
[4]偏微分方程数值解法[M].李荣华.北京,高等教育出版社.2004
[5]混合有限元基础及其应用[M].罗振东.北京,科学出版社.2006
[6]区域分解算法[M].吕涛,石济民,林振宝.北京,科学出版社.1992
[7]重调和方程的混合有限元的区域分解算法[J].王寿成,姚海波.佳木斯大学学报,1(2009), 579-581
[8]二阶椭圆问题的混合有限元分析[J].罗振东.应用数学,1992,5卷4期,26-31.
[9]三维空间的Stokes问题的混合有限元法[J].罗振东.贵州师范大学学报,1998,9卷2期,70-74.
[10]有限元混合法理论基础及其应用[J].发展与应用.山东教育出版社,济南,1996
[11] Stokes问题的混合有限元分析[J].王烈衡.计算数学,1987,9(1),70-81.
[12] Analysis of some mixed finite element methods for the Stokes problem [J].Bernardi Cand Rangel C.Math.Comput, 1985, Vol.44 No.169, 71-79.
[13] Error-bounds for finite element method [J].Babuska.I.Numer.Math.1971, Vol.16,322-33
[14] On the existence, uniqueness and approximation of Saddle point problems arising from Lagrangian multipliers [J]. RAIRO Anal. Numer.,1972,8,129-151.
[15] Mixed and hybrid Finite Element Methods [M]. F.Brezzi, M.Fortin, Springer-rerlag, 1991.
[16] A mixed Finite Element Method for the Biharmnic Equation, Symposium on Mathematical Aspects of Finite Element in Partial Differential Equations [M]. Ciarlet P G and Rariart P. C de Boor, Ed., New York:Academic press, 1974,125-143.
[17] Finite Element Methods for Navier-Stokes Equations [M]. V.Girault and P.Raviart. Berlin:Springer,1986.
[18] Schwarz算法的Lions框架与异步并行算法的收敛性证明[J].吕涛.系统科学与数学,9(2),1989,128-132.
[19]椭圆型方程区域分解方法中某些问题研究[J].储德林,博士论文,1991.
[20]关于解椭圆型问题的两个子区域不重叠区域分解算法[J].顾金生,胡显承.计算数学,4(1664),432-447.
[21]基于非协调元的区域分解算法[J],黄建国.计算数学,1(1995),47-58.
[22] On the convergence of a space decomposition methods for nonlinear problems [J].周叔子,曾金平.数学进展.28(1999),541-542.
[23]非嵌套网格上的Morley元两水平加性Schwarz方法[J].石钟慈,谢正辉.计算数学,3(
[24] On Schwarz alternating method.Ⅰ. Lions,P. Proceedings of First Internations Symposium on Domain Decomposition Methods for Patial Differential Equations, SIAM,Philadelphia,PA,1988.
[25] On Schwarz alternating method.Ⅱ. Lions,P. Domain decomposition methods, SIAM, Philadelphia,PA,1989.
[26] Two synchronous parallel algorithms for partial differential equations [J]. Lu,T., Shih,T.M. and Liem,C.B. J.Comp.Math.,9(1991),74-85.
[27]一个双调和方程的Schwarz交替法[J].蒋美群,邓庆平.计算数学,1994,(1):93-101
[28]椭圆与抛物形方程引论[M].伍卓群.北京:科学出版社, 2005,1-50.
[29] Elliptic Partial Differential Equations of Second Order [M]. D.Glibarg and N.S.Trudinger, Springer verlag, 1977.
[30]有限元逼近的离散强极值原理与Schwarz算法的收敛性[J].沈树民,蒋美群,邓庆平.高等学校计算数学学报,4(1995),339-344.
[31] Analysis of Schwarz algorithms for mixed finite element methods. Ewing,R. and Wang,J. RAIRO,MAN, 1992 (26):739-756.
[32] Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems:algorithms and numerical results, Mathew,T. Numer,math., 1993 (65):445-492