Non-Overlapping Domain Decomposition Method For Optimal Boundary Control Problems Governed By Wave Equations

Optimal boundary control problems governed by wave equations are wide-ly used in many fields, such as in weather prediction, medical science, acoustic wave and seismic wave, and so on. For two kinds of boundary control con-ditions of these problems, a non-overlapping domain decomposition method is considered in this thesis. The whole domain is divided into many non-overlapping subdomains, and the optimal boundary control problems governed by wave equation is decomposed into local problems in these subdomains. A non-overlapping domain decomposition scheme is established by using the in-tegral mean method to present an explicit flux calculation on the inter-domain boundary T. Because these local problems are mutually independent, these problems can be calculated at the same time so that it can downsize the scale of the whole problem and reduce the amount of computational work. This thesis consists of three chapters.In Chapter 1, we will introduce the backgrounds of optimal boundary control problems governed by wave equations and non-overlapping domain decomposition methods, respectively. Then, the outlines of this thesis and some of the related research work of the domain decomposition method are also presented.In Chapter 2, we will discuss the non-overlapping domain decomposition method for optimal boundary control problems governed by wave equations with Neumann boundary condition. First, we deduce the adjoint state equa-tions and the optimality condition by the theory of optimal control problem. Then, using the integral mean method, we establish the non-overlapping do-main decomposition schemes for the state equations and the adjoint state equations respectively, and the discrete optimality condition. By use of the analytical skills of a priori error estimate, we derive the optimal L2-norm error estimates. Finally, we present numerical experiments to verify the correct-ness of the theoretical results and the validity of the non-overlapping domain decomposition schemes.In Chapter 3, we will study optimal boundary control problems governed by wave equations with absorbing boundary condition. Absorbing boundary conditions is a kind of mixed boundary condition, which includes the time derivative and the space derivative of the function. Here, we consider the objective functional with final time value of both the function and the time derivation of the function. Analogously to the analysis in Chapter 2, we deduce the adjoint state equations and the optimality condition, establish the non-overlapping domain decomposition schemes for the state equations and the adjoint state equations respectively, and the discrete optimality condition. By use of the analytical skills of a priori error estimate, we derive the optimal L2-norm error estimates.