Iterative Solutions For Several Discretized Nonlinear Partial Differential Equations And Constrained Optimization Problems

Most practical problems in scientific computing and engineering applications,such as phase separation process,incompressible fluid flow problems,can be attribut-ed to the solutions of linear or nonlinear partial differential equations.Since it is difficult to compute analytic solutions or classical solutions do not even exist in some cases,numerical approximation has become the mainstream method.It has penetrat-ed into physics,chemistry,biology and other fields of modern science and engineering,and plays an important role in the development of science and technology.Using nu-merical methods to discrete problem models and transforming them into equivalent linear algebra equations are the main idea of numerical approximation.Based on d-ifferent practical problems,these linear algebra systems may have different structural characteristics,for instance,block structure or ill-conditioned properties of the coef-ficient matrices derived from different model operators,large-scale sparse structure of linear systems derived from discrete schemes and so on.How to design efficient,economical and stable numerical solution methods based on the structural charac-teristics of linear systems is a focus of modern scientific and engineering computing,which plays an crucial role in the field of numerical linear algebraThis paper is mainly concerned about fast numerical solution methods for solving linear systems discretized from three types of nonlinear partial differential equations and a type of PDE constrained optimization problem.By utilizing the structural characteristics of different discrete linear systems and adopting preconditioning tech-niques,efficient and robust iterative algorithms are designed for solving different dis-crete linear systems.The full-text consists of six chaptersIn Chapter 1,we introduce the background,significance and status of the research in detail.The main research content and innovation of this paper are illustrated at the end of the chapterChapter 2 is mainly concentrated on the numerical solution methods of linear system discretized from a nonlocal Cahn-Hilliard equation.An efficient precondition-er is proposed for the discretized block two-by-two linear system of which an indefinite matrix is contained.The main features of the proposed preconditioner are:it does not involve the indefinite matrix operations;the eigenvalues of the corresponding pre-conditioned system are all real;on the basis of having the same eigenvalues as some existing preconditioners,algorithm implementation of the proposed preconditioner only needs to solve two identical linear subsystems with symmetric positive definite matrices as coefficient,which implies the effectiveness of the preconditioner.Numer-ical experiments verify the efficiency and stability of the proposed preconditioner.Chapter 3 is mainly focus on solving the linear system arising from the con-strained optimization problem with a nonlocal Cahn-Hilliard equation as constrain-t.The block four-by-four linear system discretized by the constrained optimization problem can be transformed into an equivalent linear system with special structure by appropriate deformation.Based on the structural characteristics of the deformed coefficient matrix,a fast solver that is robust with respect to the mesh grid size and model parameters is designed to solve the discrete linear system.Eigenvalue distri-butions are discussed in detail and come to the conclusion that all eigenvalues of the preconditioned system are positive and real.In addition,eigenvalue distributions are depicted and shown that the eigenvalues of the preconditioned matrix are located in the parameter free interval[1/2,1].Finally,numerical results exhibit that the proposed preconditioner within Krylov subspace acceleration is efficient and robust for solving the discrete linear system.Chapter 4 is mainly considered numerical solution methods of the discretized linear system arising from the FitzHugh-Nagumo convection-diffusion reaction equa-tions.Starting from the linear system discretized by discontinuous Galerkin finite element method,a structure preconditioner is designed.The motivation of the pre-conditioner aims at reducing the influence of indefinite Jacobian matrix derived from the single nonlinear term on the solution of discretized linear system,and thus im-proving the computational efficiency.Algorithm implementation details show that only two subsystems with mass matrix plus stiffness matrix as coefficient matrices need to be solved.Spectral properties of the preconditioned system are analyzed.Numerical results verify the effectiveness of the proposed preconditioner.Chapter 5 is mainly aimed at fast solution methods of generalized saddle point linear systems derived from the incompressible Navier-Stokes equations.By intro-ducing a regularization matrix,a regularized-based splitting iterative method and the corresponding preconditioner are proposed.Algorithm complexity comparisons be-tween the proposed preconditioner and some existing preconditioners are discussed,which show that the regularization technique can improve the conditioning of the in-volved inner linear subsystems of iteration method.Unconditional convergence of the proposed iterative method is proved.Spectral properties of the preconditioned matrix are also given.Furthermore,a relaxed preconditioning form is presented and the spec-tral properties of the corresponding preconditioned matrix are also discussed.Finally,the effectiveness of the proposed preconditioners is verified by numerical examplesFinally,a summary of the full article and future research directions are given in Chapter 6.