| Mathematical physics and engineering problems can be turn into the problems of solving large scale partial differential equations, such as reservoir simulation[15, 34, 40,,41, 42, 42, 43, 44, 45], the design of large scale of spacecraft structure engineering, aerodynamics[46], reactor, etc. The domains they are defined on are always large area with high dimension and irregular geometry, which cause much difficulty to computation when finding their solutions. Domain decomposition methods are effective way to solve the large scale partial differential equations in parallel. These methods can decompose a large scale problem into small scale problems, making the computation more easier. Because of this, since the 50's of last century, even before the advent of the parallel computers, domain decomposition methods have been applied on sequential computer. With the development of parallel computers and parallel algorithms, since the first 80's, this kind of method has flourished. Now, super parallel computers have been used in energy industry, biology research, weather forecast and simulation, etc. Domain decomposition methods are always employed in two context. First, the division of large scale problems into several smaller subproblems by the means of domain decomposition is a way to introduce parallelism into large scale problems, and thus the computing time can be shorten. Second, many problems involve more than one mathematical model, each poses on a different domain so that domain decomposition occurs naturally, and so the parallelism is achieved. Because domain decomposition method can turn a large scale problem into smaller ones, turn complicated boundary condition into easier ones, there are very much research work on it.Under the guidance of Professor Yuan Yi-rang, the author finished this dissertation. This dissertation is some research work on nonoverlapping domain decomposition method. For several mathematical models, domain decomposition methods are proposed, as well as convergence analysis and error estimates. Several numerical experiments are also given. The whole dissertation consists of four chapters. In the first three chapters, we propose explicit/implicit domain decomposition methods to solve the parabolic equations with discontinuous coefficients, parabolic equation systems, integro-differential equations and the problem of miscible displacement in porous media, respectively. In chapter 4, we use the domain decomposition method based optimization to solve a type of parabolic equations.For every method, we all give the theoretical analysis.When we apply domain decomposition methods to solve a mathematical model, we firstly divide the domain into several subdoamins according to the features of the model or the geometry of the domain. Then we solve the subproblems independently on their own sub-domains respectively. When finding the solution of a partial differential equation, we must have known its boundary condition. However, domain decomposition is an artificial division. Then, for a subdomain, there is at least a part of boundary with unknown boundary condition, that is to say, the inner-domain boundary conditions are unknown. Our task is to give the boundary conditions to the interfaces of the subdomains or give the boundary conditions to the interfaces in the procedure though the boundary conditions can not be given apparently.The explicit/implicit domain decomposition method is a kind of method that we give the inner-domain boundary conditions explicitly. We know, if we use an explicit method, the procedure can achieve parallelism naturally. However, there is a stable constraint for the explicit method and the time step is constrained too. So, if we want to march time on, the explicit method will need more steps than implicit method. And thus, it will cost more time to find the solution. Implicit method is unconditionally stable and there is no constraint to time step. While at each time level, we must solve a large, global system of equations. When the mesh is refined, the equation systems become larger at the same tune. Solving this kind of equation system also cost much time. The explicit/implicit method include both of the advantage. It use simple, explicit calculations on the boundaries between subdomains to predict the inner domain boundary condition. Then the equation on the whole domain is decomposed into several equations on subdomains. When computing, the large, global equation system turns into several smaller ones. So the parallelism is achieved. The explicit nature of the inner-domain boundary conditions induces a time step limitation that is necessary to preserve stability, but this constraint is less severe than that which comes with a fully explicit method.There has been much work on the explicit/implicit domain decomposition methods. In Ref. [2], Dawson and his co-worker employed domain decomposition finite difference method, give the inner-domain boundary conditions explicitly and get an optimal k∞ normed error estimates. Based on this method, In Ref. [47] Du Qiang and his co-workers proposed an efficient domain decomposition finite difference method, giving the inner-domain boundary conditions by the help of solutions from the last a few levels. Comparing with Ref. [2], this procedure was more efficient. In Ref. [1], Dawson and Dupont applied a an explicit/implicit domain decomposition finite method, defining a function to predict inner-domain boundary conditions, and derive an optimal L2 normed error estimates. Then, based on this method,in Ref. [4], Dawson and Dupont used explicit/implicit domain decomposition based on block-centered finite differences and get an optimal l2 normed error estimates.In chapter 1, we consider the parabolic equation and parabolic equation system with discontinuous coefficients, parabolic equation system in three dimensions. In Ref. [1], a parabolic equation with constant coefficient was studied. While if we use its procedure to the equations with discontinuous coefficients or with variable coefficients, the advantage will be weaken. In this chapter, we propose a new kind of explicit/implicit domain decomposition method to solve the equations and get the optimal L2 normed error estimates. §1.1 is the introduction, it introduce the existing research work on parabolic equations and equation system with discontinuous coefficients and parabolic equation system. In §1.2 we consider a parabolic equation with discontinuous coefficients. We mainly study the discontinuity of the diffusive coefficient. According to this feature, we decompose the domain along the discontinuous line. Because of the discontinuity of the coefficient, the approximation to the fluxes which will be used as Neumann inner-domain boundary condition will be one order less. And thus the stable constraint, error estimates and the efficiency are all will be affected. We redefine a function to approximate the fluxes, avoiding the case of approximation order becoming less. After theoretical analysis, we get the optimal L2 normed error estimates. At the end of §1.2, we give the a numerical experiment to testify the theory. In §1.3, we consider a parabolic equation system with discontinuous coefficients. We also decompose the domain along the discontinuous line. Different from Ref. [1], which considered a constant coefficient equation, the variable coefficient equation will require a new constraint, which will affect the time step and computing efficiency. We redefine the function approximating the fluxes on the interfaces of subdonains, overcoming this difficulty. After theoretical analysis, we get the optimal L2 normed error estimate. In §1.4, we consider a type of three dimensional parabolic equation systems, proposing a domain decomposition method with characteristic finite element procedure. We have an optimal L2 normed error estimate.In chapter 2, we study the domain decomposition method for integro-differential equations. Since integro-differential equations always involve large scale problems, it is necessary to employ parallel computations. Their mathematical models are often conservative for some quality, and our methods insure the conservation. §2.1 is the introduction, which introduces the work has been done. §2.2 consider a type of integro-differential equation. Using domain decomposition moving grid finite element to solve it and derive an optimal L2 normed error estimate. In §2.3 we consider nonlinear integro-differential equations. Comparing with linear equations, the nonlinear equations introduce a problem to be solved, which is that the numerical solution can not contain the continuity of the true solution since the numerical solutionis derived from independent computation of different CPU of parallel computer. In addition, the integro term cause difficulty to theoretical analysis too. And this will cause difficulty in theoretical analysis. In §2.3, we overcome this difficulty. After the theoretical analysis, we get L2 normed error estimates. At the end of this section, we give an experiment to testify the method.In chapter 3, miscible displacement in porous media of reservoir simulation is studied. Reservoir simulation always involves large area and long period, so it needs large scale, long time computation, so it cost too much computational time. And thus, it is necessary to introduce parallelism in finding the solution of the problems. Now alternating direction methods[44, 53,54, 55] are populous parallel algorithms to be used for numerical reservoir simulation. While domain decomposition methods[48] for it are just in the incipient stage. In §3.2, we use domain decomposition method modified with characteristic procedure finite element to solve it, giving theoretical analysis and getting L2 normed error estimates. In the end of this section, we give a numerical experiment to testify the procedure. In §3.3 we unite mixed finite element with the domain decomposition method and get L2 normed error estimates.For the domain decomposition method, very important issue should be considered is the coupling of the numerical solution on the interface of subdomains. We know, numerical solution may not contain the continuity of the true solution, there will be a jump of the numerical solution. However, if the true solution is continuous, we certainly hope the numerical solution hold the continuity. So, the optimization-based domain decomposition method is proposed. This method is based on a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the inner-domain boundaries. We show that the solutions of the minimization problem exist and we derive an optimality system from which these solutions may be determined. And in each subdomain the optimization problem is constrained by the partial differential equations. Some research work has been done. In [36], optimization based nonoverlapping domain decomposition algorithms were presented. In [50], a type of elliptic equation was studied, and H1 normed error estimates has derived.In chapter 4, we use the optimization-based domain decomposition method to solve a type of parabolic equations. For this type of parabolic equations, there are not much work on it[49, 52] .Ref.[49]gave an optimization-based domain decomposition by using the adjoint state equations , while there was no theoretical analysis. In Ref. [52], though convergence analysis and numerical experiments were given, there was no error estimate. But this typeof parabolic equations are used widely, so it is necessary to find a parallel procedure for it. In chapter 4, we define an objective functional, which is composed of the jump of numerical solution and a penalty term. The subproblems of (4.1.1) on subdomains is regarded as the constraints to the objective functional. We can prove that when the penalty term turns to zero, the optimal solution of the objective functional is the solution of (4.1.1). §4.1 is the introduction, it gives the objective functional. In §4.2 we prove the existence of the solution. In §4.3, we give the convergence of the optimal solution to the solution of (4.1.1). In §4.4, the optimal system and its constraints are presented. In §4.5, the finite element method is lay out, and we get the optimal H1 normed error estimate by the means of theoretical analysis.
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