The Research On Domain Decomposition Methods For Two Kinds Of Nonlinear Systems

Many problems arised in science and engineering are often large scale, and require high precision. Hence, we need to design some new and more efficient algo-rithms for such problems. With the use of parallel computer, it is a very important way to solve those problems in parallel. Meanwhile, as the technology and size of parallel computing systems advance, so does the detail at which we can solve very complicated numerical problems, including optimization problems constrained by nonlinear partial equations (PDEs) and complementarity problems including, for example, obstacle problems and free boundary value problems. This trend of in-creasing computational complexity demands both the design of scalable parallel numerical algorithms and the adoption of modern software engineering techniques for the development of numerical libraries. Domain decomposition method was developed in 1980s. Its main point is to divide the domain into several small sub-domains, and to solve the subproblems in related subdomains. Domain decomposi-tion methods are widely used and very powerful for solving large sparse linear and nonlinear systems of equations arising from partial differential equations (PDEs). Among different families of domain decomposition methods, we focus primarily on the class of Schwarz type methods. This dissertation proposes and tests some gen-eral techniques of the linear preconditioning based on a Schwarz framework for two such challenging problems, namely boundary control of unsteady incompressible flows in computational fluid dynamics and nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems.This thesis presents the development of robust, scalable and parallel numerical methods for these two kinds of nonlinear systems, and consists of the following three parts.First, we consider an algebraic multiplicative Schwarz iteration scheme for solving the linear complementarity problem that involves an H+-matrix. We show that the sequence generated by the multiplicative Schwarz iteration scheme con-verges to the unique solution of the problem without any restriction on the initial point. For different overlapping sizes, the convergence rate of the proposed method is analyzed in an algebraic setting. Moreover, we establish monotone convergence of the proposed method under appropriate conditions. Numerical results show that efficiency can be achieved by the multiplicative Schwarz iterationIn the following, we develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of Newton-Krylov-Schwarz meth-ods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly indepen-dent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems.Finally, we propose and investigate a class of parallel full space SQP Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithms for some time-dependent boundary control problems of two-dimensional incompressible Navier-Stokes equations, which is fully implicit and allows large time steps. In LNKSz, a Lagrangian functional is formed and differentiated to obtain a Karush-Kuhn-Tucker (KKT) system of nonlinear equations. Inexact Newton method with line search is then applied. At each Newton iteration the linearized KKT system is solved with a Schwarz pre-conditioned Krylov subspace method. We show that LNKSz is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large scale calculations involving several millions unknowns obtained on machines with more than one thousand processors.