Preconditioned Methods For The Fractional Diffusion Equations Discretized By Mixed Finite Element Method

In recent decades,the research results of fractional diffusion equations have been widely used in many fields.The analytical methods of solving fractional diffusion equations,including Mellin transform,Laplace transform and Fourier transform,are only suitable for solving some simple or special cases.For general fractional diffusion equations,numerical methods are more often used.Because of the nonlocality of fractional differential operators,the discrete coefficient matrices are often dense matrices which bring many difficulties to numerical calculations.Therefore,it is necessary to find fast numerical algorithms to solve fractional diffusion equations.In this thesis,we mainly study the preconditioned methods for the fractional diffusion equations with variable coefficients.By introducing an intermediate variable and then utilizing the mixed finite element method,the original problem can be transformed into two groups of linear systems.As the coefficient matrix of the first group is tridiagonal,the first group can be solved efficiently by the direct method.Although the second linear system is dense and usually ill-conditioned,it has some special structure.Our goal is to construct fast preconditioned iterative methods for the second linear system.The main results of this thesis are as follows:(1)Based on the special structure of the problem,we rearrange the columns and rows so that the new coefficient matrix can be written into a 2×2 block matrix with one diagonal block being very small and another being the Toeplitz structure.Then we propose the block diagonal preconditioner and block triangular preconditioner and analyze the eigenvalue distribution of the preconditioned matrices.(2)In order to reduce the computational cost,we approximate the Toeplitz matrix with certain circulant matrix in the preconditioner.The properties of the new precondi-tioners for both the one-sided fractional diffusion equation and two-sided fractional diffusion equation are studied in detail.(3)Numerical tests are carried out to illustrate the performance of our preconditioners.