【作者】 彭大萍；

【导师】 丁睿；

【作者基本信息】 苏州大学 ， 应用数学， 2004， 硕士

【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。更多还原

【Abstract】 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.更多还原