The Research of Domain Decomposition Methods for Parabolic Variational Inequalities
|Keywords||parabolic variational inequalities domain decomposition method finiteelement discretization optimization problem convergence analysis|
The numerical method of domain decomposition method is new and efficient forsolving the partial differential equation, it gains excellent performance in parallel. Basedon the idea of “divide and rule”, the original large and complicated domain can be reducedin each sub-domain, so that the original problems can be analyzed in each sub-domain. Atpresent, the domain decomposition method in solving the variational inequalities is mainlyconfined to the elliptic variational inequalities while there is a few application in solvingthe parabolic variational inequalities. In this paper, a kind of domain decompositionmethod is proposed to solve the parabolic variational inequalities.Firstly, this paper introduces the research progress and development status about thevariational inequalities and domain decomposition method for solving the variationalinequalities. Then, some theoretical knowledge of the variational inequalities and somealgorithm of domain decomposition method are presented, especially elaborates thenecessity of this subject.Secondly, we construct domain decomposition algorithm for the solution of thesecond type parabolic variational inequalities on the background of time dependentfriction problem in mechanics. Using semi-discretization and implicit method in timeformulated the parabolic variational inequalities as an elliptic variational inequality, thenwe adopt the numerical integrals to approximate the non-differentiable term which cannotcalculus easily, and the calculation becomes more convenient. Based on the equivalentoptimization problem, the domain decomposition method was proposed and itsconvergence was also given, at the same time, the numerical example illustrates thefeasibility and effectiveness of the method. Then, we construct domain decompositionalgorithm for the solution of the first type parabolic variational inequalities, theconvergence of the corresponding algorithm was given. Also there is a numerical examplebe given to illustrate the feasibility and effectiveness of the method.Lastly, this paper extends domain decomposition method to solve a kind of fourthorder parabolic variational inequalities with obstacle constraint, and constructs domain decomposition algorithm for the solution of the variational inequalities. Moreover, thenumerical example is given to prove that the method is flexible.