An Adaptive Parallel AMG Method For A Class Of Space-time Caputo-riesz Fractional Diffusion Problems

Fractional diffusion equation is a widely used class of fractional differential equation. In this thesis, for a class of space-time Caputo-Riesz fractional diffusion problems, we construct a fully discrete scheme of m-order finite element method in temporal direction and linear finite element method in spatial direction, respectively.Next we prove that the coefficient matrix of the scheme is a block Toeplitz matrix on a uniform spatial mesh. The numerical experiments show that the fully discrete finite element solution function possesses the saturation error order under L2(?)norm. For the fully discrete system of space-time finite element scheme, we obtain the estimation on the condition number of the coefficient matrix of discrete system via numerous numerical experiments. Based on the estimation and fast Fourier transform, we construct and analyse an adaptive parallel algebraic multigrid method(AMG) with low complexity for solving the fully discrete finite element scheme.The numerical results show that the new algorithm is robust,efficient and has good parallel speedup. Furthermore, we give out a reference formula on the strength threshold which affect the robustness of AMG method, and construct an AMG method with higher efficiency based on the adaptive selection of the threshold.