Research On Effective Preconditioning Methods For Solving Linear Systems With Three-by-Three Structure

Numerical solutions of partial differential equations play a significant role in many fields of engineering application and computational science,such as fluid dynamics,constrained quadratic programming,optimal control problems,reservoir simulation experiments and power and energy systems,and the above problems can usually be attributed to the solution of large-scale sparse linear systems through numerical discretization.The coefficient matrices of these linear systems are usually characterized by asymmetry,strong indefiniteness and poor spectral properties.How to solve such problems efficiently is a hot topic in this discipline and related fields,which has important significance for practical applications.In this paper,we mainly study the efficient numerical solution of large sparse linear systems with three-by-three block structure,and focus on preconditioning technology to accelerate Krylov subspace methods.According to the special structure of the coefficient matrices of linear systems,several kinds of efficient preconditioning methods are proposed by means of dimension expanded technology,shift-splitting technology and parameterized technology,and the corresponding algorithm complexity and convergence analysis are studied in detail.Finally,the effectiveness and feasibility of the proposed preconditioners are verified through numerical experiments.