Preconditioners For Boundary Value Methods Discretizations Of Several Kinds Of Space Fractional Diffusion Equations

During recent years,fractional diffusion equations(FDEs),and their applications to modeling anomalous diffusion phenomena are widely recognized.And it has been applied in more and more fields,including mechanical and electrical engineering,turbulent flow,biology,finance,image processing and data fitting.Since the analytical solutions of fractional diffusion equation is difficult to obtain,the numerical method of FDEs has become the main solution method and has been widely developed and studied,so numerical solutions for FDEs become main ways and then have been developed intensively,such as finite difference method,finite element method,discontinuous Galerkin method and other numerical methods.The boundary value method(BVMs)is a new method for solving ordinary differential equations based on the linear multistep formulas,which overcomes the limitation of linear multistep formulas.At present,the boundary value method has been successfully applied to differential and integral-differential equations.BVMs have been applied successfully to a wide range of differential and integrodifferential equations.Due to the nonlocal nature of the fractional differential operator,numerical solution of FDEs usually represents very intensive computational task.For the discrete linear system,the direct calculation is more complicated and the convergence rate is slower.The purpose of this paper is to construct a fast preconditioning iterative algorithm for BVMs discrete system of space fractional diffusion equation.The high-order method is used to discretize the fractional diffusion equation.We construct the preconditioner based on Kronecker product splitting,and use it to accelerate the GMRES iterative algorithm.It can accelerate the iteration speed and improve the computational efficiency,which has important academic and application significance.The main contents of this paper are as follows:In the first part,we concerned with the construction of efficient preconditioners for systems arising from boundary value methods time discretization of space fractional diffusion equations.The boundary value methods lead to a coupled block system which is in the form of the sum of two Kronecker products.Our approach is based on an alternating Kronecker product splitting technique which leads to a splitting iteration method.We show that the splitting iteration converges to the unique solution of the linear system and derive the optimal values of the involved iteration parameters.The splitting iteration is then accelerated by a Krylov subspace method.One component of the Kronecker product preconditioners has the same structure as the matrix derived from implicit Euler discretization of the problem.We reuse the structure preserving approximation as the building block for our KPS preconditioner.Finally,several numerical experiments are presented to show the effectiveness of this method.In the second part,a distributed optimal control problem with the constraint of a fractional differential equation is considered.Boundary value methods are used to solve the coupled initial and final value problems arising from the first order optimality conditions for this problem.We use a GMRES method with Kronecker product splitting preconditioning method for solving the resulting two-by-two linear system.The preconditioner is a block Kronecker product structure.We obtain this Kronecker product through an alternating Kronecker product splitting iteration method,and prove the convergence of this method.Numerical experiments are presented to illustrate the accuracy and computational efficiency of the proposed approach.