Stability of Space-Time Finite Element Discretizations of Subdiffusive Time-Fractional Differential Equations

The study of anamolous diffusion, and in particular of subdiffusive time-fractional differential equations, is of great interest for its ability to describe transport of particles through porous media. The time-fractional derivatives in these problems are nonlocal, which notably hinders performance of classical time-stepping methods. As a result, there is significant interest in using simultaneous space-time discretizations for these subdiffusive problems, which have the additional benefit of significantly relaxing the regularity requirements for candidate solutions. These types of Petrov-Galerkin schemes require a careful choice of discretized trial and test spaces in order to guarantee stability; in particular, the usual Galerkin choice $U^h = V^h$ is generally unstable. We propose a stable, nonadaptive Petrov-Galerkin discretization of a model subdiffusive problem. We show how stable pairs of discretized trial and test spaces can be constructed by tensorizing temporal and spatial trial and test spaces with known good properties. We then demonstrate that the discretized Petrov-Galerkin problem can be reformulated quasi-optimally as a minimal functional residual problem, which can be numerically solved by solving an equivalent algebraic residual minimization problem.