Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems

TL;DR

This paper introduces a novel space-time Petrov-Galerkin finite element method for solving time-fractional diffusion equations with Riemann-Liouville derivatives, providing theoretical analysis and numerical validation.

Contribution

It develops a new finite element formulation for fractional diffusion problems, proving well-posedness, stability, and error bounds, with extensive numerical verification.

Findings

The method is well-posed with proven inf-sup stability.

Error bounds are established in energy and L2 norms.

Numerical examples confirm convergence and accuracy.

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 $\alpha\in(0,1)$ 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 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.