Space-Time Petrov–Galerkin FEM for Fractional Diffusion Problems

Abstract

Abstract 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 ∈ ( 0 , 1 ) {\alpha\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 derive error bounds in both energy and L 2 {L^{2}} norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L 2 {L^{2}} stability property of the L 2 {L^{2}} projection operator plays a key role. We provide extensive numerical examples to verify the convergence analysis.