Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems
It provides a rigorous numerical framework for fractional diffusion problems, which are important in modeling anomalous diffusion, but the contribution is incremental as it extends existing Petrov-Galerkin methods to a specific fractional operator.
The paper develops a space-time Petrov-Galerkin finite element method for time-fractional diffusion equations with Riemann-Liouville derivatives, proving well-posedness and deriving error bounds in energy and L2 norms, with numerical verification of convergence.
We present and analyze a space-time Petrov-Galerkin finite element method for a time-fractional diffusion equation involving a Riemann-Liouville fractional derivative of order $α\in(0,1)$ in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness. Further, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and establish error bounds in both energy and $L^2$ norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard $L^2$ stability property of the $L^2$ projection operator plays a key role. We provide extensive numerical examples to verify the convergence of the method.