Research On Theoretical And Numerical Methods Of Obstacle Problems

The obstacle problem is a typical variational inequality of the first kind,which appears in the fields of physics,financial management science,engineering applications and so on.This problem has attracted much attention from scholars because of its inherent mathematical theory and wide applications.After decades of research and development,systematic mathematical theoretical analysis and many effective numerical algorithms have been formed.Related issues,such as the optimal control problem governed by obstacle problems,have also received extensive attention.This dissertation focuses on the subject of “obstacle problems”.Numerical methods of a unilateral obstacle problem,a bilateral obstacle problem and an optimal control problem constrained by an obstacle problem are studied,respectively.At the same time,a new type of obstacle problem is proposed.The corresponding theoretical and numerical analysis are presented.The main work is listed as follows:(1)Recently,more and more evidences show that second-order in time dissipative systems have remarkable optimization properties.In Chapter 2,we consider applying the related dynamic functional particle method to calculate the unilateral obstacle problem.First,we construct an equivalent format of the obstacle problem which transforms the underlying problem into a nonlinear equation.Then,the dynamic functional particle method is applied to solve the nonlinear equation.The experimental results show that our algorithm converges as fast as the penalization method under the condition of more flexible selection of parameter.And our algorithm barely depends on initial values and the mesh size.(2)In Chapter 3,we consider a new type of numerical algorithm for the bilateral obstacle problem.Firstly,by introducing a parameter,the complementary form of the discrete bilateral obstacle problem is equivalently rewritten into a non-differentiable nonlinear equation system that is easier to solve.Secondly,the obtained nonsmooth equation is regularized,and then the Newton iteration method is used to numerically solve the nonlinear equation system.Experimental results show that the algorithm can quickly calculate the numerical solution.The rate of convergence does not depend on the mesh size.(3)In Chapter 4,we continue to apply the dynamic functional particle method to solve an optimal control problem constrained by an obstacle problem.Firstly,an approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a penalty method,and the corresponding first-order optimality system is deduced.The connection between the two optimal control problems is established through some convergence results.Then,a sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum.Finally,the dynamic functional particle method is developed to solve the optimality system.Several numerical experiments illustrate the effectiveness of the proposed method.The experimental results also show that our algorithm converges fast and does not depend on the initial data.(4)To the best of our knowledge,the obstacle in most related references is perfectly rigid.In Chapter 5,we study a new type of obstacle problem,called elasticrigid obstacle problem,considering the contact between the elastic membrane and a deformable obstacle.The elastic membrane is constrained to lie above an obstacle which is made of a rigid body covered by a layer made of soft material.The soft layer is deformable and allows penetration.Under certain assumptions,we deduced three equivalent descriptions of the problem,and proved its existence and uniqueness of the solution.Then,based on the variational inequality form,we derive an optimal order error estimate for the finite element approximate solution under appropriate solution regularity assumptions.Finally,numerical calculations are carried out combined with the penalty method,and the simulation results are in good agreement with the theoretical analysis.