Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems

摘要

We present and analyze a space-time Petrov-Galerkin finite element method fora time-fractional diffusion equation involving a Riemann-Liouville fractionalderivative of order $\alpha\in(0,1)$ in time and zero initial data. We derive aproper weak formulation involving different solution and test spaces and showthe inf-sup condition for the bilinear form and thus its well-posedness.Further, we develop a novel finite element formulation, show the well-posednessof 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-supcondition, a certain nonstandard $L^2$ stability property of the $L^2$projection operator plays a key role. We provide extensive numerical examplesto verify the convergence of the method.