Efficient Preconditioners For Several Classes Of Discrete Systems With Periodic Structure Coefficient Matrices

The radiation diffusion equation is the basic model describing the radiation transmission process.The symmetric finite element method is one of the most commonly used discrete methods for numerically solving the equation.one of the most commonly used discrete methods to solve the equation numerically.But in solving these kinds of problems,The large scale of discrete systems,periodic boundary conditions and large deformed meshes are challenges,resulting in poor condition number of its discrete system,Therefore,it is necessary to design an efficient preconditioning algorithm.In this paper,the efficient preconditioned algorithm for discrete systems for solving the periodic structural coefficient matrix of two cases is studied.Firstly,for the discrete system of the two-dimensional radiation diffusion problem of the periodic structure coefficient matrix in the first case,A new approximate chasing method Preconditioner is designed by introducing reasonable weight coefficients.the one-dimensional sub-problem of the inner cycle is solved by the Shur complement method with chasing method.Then,for the discrete system of the threedimensional radiation diffusion problem of the periodic structure coefficient matrix in the first case,By introducing the surface roughening strategy and two kinds of three-dimensional block polishing operators,a multi-grid method preconditioner based on block(face)roughening is designed.On this basis,for the characteristics of the discrete system of the periodic structure coefficient matrix in the second case,the above-mentioned multi-layer grid method based on block(face)roughening.The restriction operator and polishing operator are improved,and a suitable method for solving periodic boundary conditions is designed.A new multi-grid method preconditioner for the three-dimensional radiation diffusion problem.Finally,the corresponding preconditioned conjugate gradient method solver is designed for the above three preconditioners.Numerical experiments show that these preconditioned conjugate gradient method solvers are robust and efficient.