第二类抛物型变分不等式的边界元近似
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摘要
本文讨论了含有不可微项的第二类抛物型变分不等式的边界元近似。首先采用时间项半离散和隐格式方法将抛物型变分不等式化解为一个含有不可微项的第二类椭圆型变分不等式，给出了相应的等价单侧边值问题，并讨论了非线性不可微的混合变分不等式解的存在唯一性，然后引入常规边界元方法和MRM-边界元方法。对于常规边界元方法通过引入齐次Helmholtz方程建立边界积分方程，并利用边界积分方程把区域型第二类混合变分不等式转化为相应的边界混合变分不等式，分析了其解的存在唯一性；对于MRM-边界元方法则是采用MRM(多重互易方法)方法将等价单侧边值问题化解为MRM-边界混合变分不等式，并给出了MRM-边界混合变分不等式解的存在唯一性。采用正则化方法处理后给出了该边界混合变分不等式的两种数值方法。一是利用等价非线性变分方程给出了迭代方法讨论了迭代解的收敛性。另一个是通过引入变量将原边界混合变分不等式化解为标准的凸极值问题，利用标准凸极值方法可以求解。最后给出了解的误差估计。为使用边界元方法数值求解提供了理论依据。
In this paper, it discuss the boundary elerient approximation of the parabolic variational inequalities with non-differentiable friction of the second kind. First, the parabolic variational inequalities of the second kind can be turned into an elliptic variational inequality by using semi-discretization and implicit method in time,its corresponding equalent unilateral boundary problem are given; then the existence and uniqueness for the solution of nonlinear indifferenliable mixed variational inequality are discussed. Then the normal boundary element method and MRM-boundary element method are introduced respectively. For the normal boundary element method, by introducing homogeneous Helmholtz equation, it establish a boundary integral equation which can turn the domain problem into its corresponding boundary mixed variational inequality. The existense and uniquess of the solution are obtained. For the MRM-method , it turn the unilateral boundary problem into an MRM-boundary mixed variational inequality. Its corresponding boundary variational inequality and the existence and uniqueness are given. Applying regularization, two kinds of numerical methods for the regularized problem are presented. First is the convergence of the iterative method for its equivalent nonlinear boundary variational equation. Second, introducing the transformation, the mixed variational inequality is reduced to a standard convex optimization problem, it can be solved by any standard methods of convex optimization. Finally, the error estimation of the solution are given. This provides the theoretical basis for using boundary element method to solve the mixed variational ineqv ality.
In this paper, it discuss the boundary elerient approximation of the parabolic variational inequalities with non-differentiable friction of the second kind. First, the parabolic variational inequalities of the second kind can be turned into an elliptic variational inequality by using semi-discretization and implicit method in time,its corresponding equalent unilateral boundary problem are given; then the existence and uniqueness for the solution of nonlinear indifferenliable mixed variational inequality are discussed. Then the normal boundary element method and MRM-boundary element method are introduced respectively. For the normal boundary element method, by introducing homogeneous Helmholtz equation, it establish a boundary integral equation which can turn the domain problem into its corresponding boundary mixed variational inequality. The existense and uniquess of the solution are obtained. For the MRM-method , it turn the unilateral boundary problem into an MRM-boundary mixed variational inequality. Its corresponding boundary variational inequality and the existence and uniqueness are given. Applying regularization, two kinds of numerical methods for the regularized problem are presented. First is the convergence of the iterative method for its equivalent nonlinear boundary variational equation. Second, introducing the transformation, the mixed variational inequality is reduced to a standard convex optimization problem, it can be solved by any standard methods of convex optimization. Finally, the error estimation of the solution are given. This provides the theoretical basis for using boundary element method to solve the mixed variational ineqv ality.
引文
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[2] Weimin Han,Mircea Sofonea, Evolutionary Variational Inequalities Arising in Viscoelastic Contact Problems[J],SIAM Journal on Numerical Analysis,2000,38(2):556-579
[3] JiuHua Chen,Weimin Han and Mircea Sofonea,Numerical Analysis of a Nonliear Evolutionary System with Applications in viscoplasticity[J],SIAM J.Numer.Anal,2000, 38(4): 1171-1199
[4] A.Czekanski,S.A.Meguid,Analysis of dynamic frictional Contact problems using variational inequalities[J],Finite Elements in Analysis and Design,2001,37:861-879
[5] Glowinski R.,Numerical Methods for Nonlinear Variational Problems[M], New York: Springer-Verlag, 1984
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[8] Ding Fangyun, Zhang Xin, Ding Rui. Boundary mixed variational inequality in friction problem[J], Applied Mathematics and Mechanics, 1999,20(2):201-210
[9] 彭大萍，丁睿，第二类抛物型变分不等式的边界元近似，甘肃科学学报，2004， 16(1)：1-6
[10] 彭大萍，丁睿，第二类抛物型变分不等式中的MRM-边界混合变分不等式，力学季刊，已接收。
[11] Brezis,H.,Problemes unilateraux[J], Journal de Mathematiques Pures et Appliquees, 1972,51(1):1-168
[12] Han, H., A direct boundary element for Signorini problems[J], Math. Compute,1990, 55:115-128
[13] V.Sladek, J.Sladek, M.Tanaka, Boundary element solution of some structure-acoustic coupling problems using the Multiple Reciprocity Method[J], Comm. Numer. Method. Eng., 1994, 10 (2): 237-248
[14] N.Kamika, E.Andoh, A note on multiple reciprocity integral formulation for the Helmholtz equation [J], Commun. Numer. Methods, Eng., 1993, 9:9-13
[15] 祝家麟，椭圆边值问题的边界元分析[M]，北京：科学出版社，1991
[16] 丁睿，分析中的现代计算方法[M]，四川：四川科学技术出版社，2003
[17] 李挺，丁睿，粘弹性薄板动力响应问题MRM方法及收敛性分析[J]，苏州大学学报，2002，18(2)：15-22
[18] 丁方允，丁睿，一类椭圆型方程边值问题的边界积分方法[J]，兰州大学学报，1999，35(4)：25-32．
[19] Han,H., The boundary integro-differential equations of elliptic boundary value and their numerical solution, Sci. Sinica series A, 1988,29(10):1153-1165
[20] Kornhuber, R., Monotone multigrid methods for elliptic variational inequality, Numer. Math. 1996,72(2):481-499
[21] Atkinson, K., Han, W., Theoretical Numerical Analysis-A Functional Analysis Framework [M], New York:Springer-Verlag, 2001
[22] Babuska I,Aziz, AK.Survey on the mathematical foundations of the finite element method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,Aziz AK(ed.). New York:Academic Press, 1972:5-359
[23] Claes Johnson, A convergence estimate for an approximation of a parabolic variational inequality, SIAM J.Numer.Anal, 1976,13(4): 599-606
[24] 邓建中，刘之行，计算方法[M]，陕西：西安交通大学出版社，2001
[2] Weimin Han,Mircea Sofonea, Evolutionary Variational Inequalities Arising in Viscoelastic Contact Problems[J],SIAM Journal on Numerical Analysis,2000,38(2):556-579
[3] JiuHua Chen,Weimin Han and Mircea Sofonea,Numerical Analysis of a Nonliear Evolutionary System with Applications in viscoplasticity[J],SIAM J.Numer.Anal,2000, 38(4): 1171-1199
[4] A.Czekanski,S.A.Meguid,Analysis of dynamic frictional Contact problems using variational inequalities[J],Finite Elements in Analysis and Design,2001,37:861-879
[5] Glowinski R.,Numerical Methods for Nonlinear Variational Problems[M], New York: Springer-Verlag, 1984
[6] Duvant G, Lions J L. Les Inequations en Mecanique et en Physique [M], Dunod, 1972
[7] Mosco U., Approximation of the solutions of some variational inequalities[J], Ann.scuola Normale Sup. Pisa, 1967,21:373-394
[8] Ding Fangyun, Zhang Xin, Ding Rui. Boundary mixed variational inequality in friction problem[J], Applied Mathematics and Mechanics, 1999,20(2):201-210
[9] 彭大萍，丁睿，第二类抛物型变分不等式的边界元近似，甘肃科学学报，2004， 16(1)：1-6
[10] 彭大萍，丁睿，第二类抛物型变分不等式中的MRM-边界混合变分不等式，力学季刊，已接收。
[11] Brezis,H.,Problemes unilateraux[J], Journal de Mathematiques Pures et Appliquees, 1972,51(1):1-168
[12] Han, H., A direct boundary element for Signorini problems[J], Math. Compute,1990, 55:115-128
[13] V.Sladek, J.Sladek, M.Tanaka, Boundary element solution of some structure-acoustic coupling problems using the Multiple Reciprocity Method[J], Comm. Numer. Method. Eng., 1994, 10 (2): 237-248
[14] N.Kamika, E.Andoh, A note on multiple reciprocity integral formulation for the Helmholtz equation [J], Commun. Numer. Methods, Eng., 1993, 9:9-13
[15] 祝家麟，椭圆边值问题的边界元分析[M]，北京：科学出版社，1991
[16] 丁睿，分析中的现代计算方法[M]，四川：四川科学技术出版社，2003
[17] 李挺，丁睿，粘弹性薄板动力响应问题MRM方法及收敛性分析[J]，苏州大学学报，2002，18(2)：15-22
[18] 丁方允，丁睿，一类椭圆型方程边值问题的边界积分方法[J]，兰州大学学报，1999，35(4)：25-32．
[19] Han,H., The boundary integro-differential equations of elliptic boundary value and their numerical solution, Sci. Sinica series A, 1988,29(10):1153-1165
[20] Kornhuber, R., Monotone multigrid methods for elliptic variational inequality, Numer. Math. 1996,72(2):481-499
[21] Atkinson, K., Han, W., Theoretical Numerical Analysis-A Functional Analysis Framework [M], New York:Springer-Verlag, 2001
[22] Babuska I,Aziz, AK.Survey on the mathematical foundations of the finite element method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,Aziz AK(ed.). New York:Academic Press, 1972:5-359
[23] Claes Johnson, A convergence estimate for an approximation of a parabolic variational inequality, SIAM J.Numer.Anal, 1976,13(4): 599-606
[24] 邓建中，刘之行，计算方法[M]，陕西：西安交通大学出版社，2001