Fast Preconditioned Iterative Methods For Fractional Differential Equations On Locally Refined Composite Mesh

The fractional differential operator have a very wide range of applications in en-gineering and scientific computations.One of the main characteristics of the fractional differential operator is nonlocality.Numerical methods for fractional diffusion equa-tions usually generate full coefficient matrices,which makes numerical computation very difficult and complicated.Moreover,solutions of fractional differential equations may exhibit boundary layer and poor regularity even if the diffusivity coefficient and right-hand side are smooth.In this thesis,we consider the preconditioned iterative methods for the fractional differential equations whose solution exhibit boundary layer.A locally refined compos-ite mesh is employed to resolve boundary layers of the solution.The Toeplitz structure of the resulted linear systems is destroyed because of the nonuniform mesh.However,we find that the the stiffness matrix can be decomposed as a product of a diagonal matrix and a 3-by-3 block matrix,whose diagonal blocks have certain Toeplitz struc-ture.Based on this observation,we develop a block diagonal preconditioner and a block lower-triangular preconditioner.Both theoretical analysis and numerical tests show that the new preconditioners are very efficient.We also consider the local mesh refinement.In order to reduce the computational cost,we introduce an acceleration composite mesh so that the point number of locally refined mesh for the numerical solution to achieve the required accuracy is as small as possible.Meanwhile,we also develop a balanced composite mesh so that the numerical solution can achieve more higher accuracy while keeping the size of uniform mesh unchanged.Numerical experiments are carried out to test the performance of these two new composite mesh.