An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation

In this paper, we design and analyze a space-time spectral method for the subdiffusion equation. Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable psi-fractional Sobolev spaces and the new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials (GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.