An Efficient Space-Time Method for Time Fractional Diffusion Equation

A space-time Petrov-Galerkin spectral method for time fractional diffusion equations is developed in this paper. The Petrov-Galerkin method is used to simplify the computation of stiffness matrix but leads to full non-symmetric mass matrix. However, the matrix decomposition method based on eigen-decomposition is numerically unstable for non-symmetric linear systems. A QZ decomposition is adopted instead of eigen-decomposition. The QZ decomposition has essentially the same computational complexity as the eigen-decomposition but is numerically stable. Moreover, the enriched Petrov-Galerkin method is developed to resolve the weak singularity at the initial time. We also carry out the error analysis for the proposed methods and present ample numerical results to validate the accuracy and robustness of our numerical schemes.