Block Splitting Preconditioning Method For Structural Linear Systems In Buoyancy Driven Flow Problems

Many problems in large-scale scientific and engineering calculations,such as fluid dynamics,optimal control problems,elastic mechanics,etc.,can be abstracted into the problem of solving partial differential equations through mathematical modeling.Since it is difficult to compute analytic solutions,numerical methods have become the common method,and plays an important role in dealing with many sophisticated problems in the field of modern science and engineering.With appropriate numerical discretization,these problems can be transformed into linear algebra systems within some certain structures.Generally,these linear systems possess special structure,such as 2 × 2 block or 3 × 3 block structure,and are usually large,sparse,and ill-conditioned.Designing efficient,robust and economic numerical algorithms based on the special structure of the linear system is not only the core of modern scientific and engineering computing,but also a research hotspot for current numerical computing workers.This paper is mainly concerned about fast numerical solution methods of structural linear systems arising from buoyancy-driven flow problems.By utilizing the structural characteristics of the discrete linear systems and adopting preconditioning techniques,efficient and robust iterative algorithms are designed for solving the discrete linear systems.The full-text consists of four chapters:In Chapter 1,we introduce the background,significance and status of the research in detail.The main research content and organization of this paper are illustrated at the end of the chapter.Chapter 2 is mainly concentrated on the preconditioned iterative methods for the linear system arising from the unsteady Rayleigh-Bénard convection problem.This chapter consists of two parts:（i）A class of efficient block-splitting preconditioner and its relaxed variants are proposed for the discretized block three-by-three linear system,which is discretized by the Newton’s linearization scheme and finite element method.Compared with the existing methods,the proposed preconditioners do not involve the explicit Schur complement matrix operation,and the algorithm implementations only need to solve two linear subsystems with positive definite matrices as coefficient matrix.Spectral distributions of the corresponding preconditioned systems are discussed.（ii）This part is mainly focus on solving the linear system arising from the unsteady Rayleigh-Bénard convection problem,which is discretized by the Picard’s linearization scheme and finite element method.Using the positive definite properties of the sub-matrix blocks in the coefficient matrix,and based on matrix-splitting idea,a class of asymmetric matrix splitting iterative methods and the corresponding splitting preconditioner are proposed to solve the discrete linear system.The algorithm complexity comparison of the proposed preconditioners are given,which show that the proposed methods can improve the computing efficiency to a certain extent.In addition,the convergence of the proposed iterative method are given by using the properties of discrete matrices.In Chapter 3,numerical experiments are listed to show that the proposed preconditioners within Krylov subspace acceleration are efficient for solving the discrete linear system in terms of iteration step and computing time.Finally,a summary of the full article is given in Chapter 4.