The Boundary Element Approximation Of The Parabolic Variational Inequalities Of The Second Kind

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.