Projection And Improved Projection Methods For Solving Free Boundary Problems

The free boundary problem is also known as the obstacle problem,which play an important.role in the engineering fields,such as the seepage problem,the free dam prob-lem and double elastic membrane problem.These problems are different from the general differential boundary value problem.They are described by some complementary forms consisting of some differential equations and inequalities which make them strongly non-linear.For the free boundary problem,it is very difficult in mathematical modelling,theoretical analysis and numerical computation.It is well known that the variational inequality is an important tool to study various free boundary problems.The basic idea is to use the Green formula to transform the problem into a va.riational inequality in the domain,and the existence and uniqueness of the solution are established by using the Lax-Milgram theorem and Sobolev space theory.Because of the inherent nonlinearity,the accurate and efficient numerical simulation of contact problems is still a very active field of research.This dissertation consists mainly of two parts.The first part concerns the projection method of the free boundary problem.The method uses the central difference scheme to discrete free boundary problem as a linear complementarity problem,which is equivalent to a fixed point problem.Then our projection method is proposed for the free boundary problem.This method only need to solve a system of linear equations for each iteration,and its coefficient matrix is fixed.The convergence of the method is proved by using the positive definite property of the matrix and projection properties.The second part is devoted to the modified projection method by using self-adaptive rule for the parameter.The convergence speed of the projection method is greatly influenced by the value of parameter which is difficult to choose for individual problems.To improve the performance of this method,we present a self-adaptive rule based on the iterative data to adjust the parameter automatically.In this way,the method has a fast and stable convergence speed for any positive parameter.And the convergence of the method is proved.Finally,some numerical results show the accuracy and effectiveness of the projec-tion and improved projection methods and we also compare our results with the known numerical results in other references.