A Petrov-Galerkin finite element method for fractional convection-diffusion equations (Q2791762)

The authors consider the numerical approximation, using the Petrov-Galerkin finite element method, of one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order \(\alpha \in (3/2,2)\) in the leading term and both convection and potential terms. They first develop variational formulations of Petrov-Galerkin type for the continuous problem along with a proof of the well-posedness of the formulations and sharp regularity pickup of the variational solutions. A new finite element method is developed, which uses continuous piecewise linear finite elements and ``shifted'' fractional powers for the trial and test space, respectively. Optimal error estimates in \(L^2\) and \(H^1\) norms are derived. One of the main feature the numerical method is that on a uniform mesh, the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned. In the cae of Riemann-Liouville, an enriched finite element method is suggested to improve the convergence. Several numerical experiments are presented to support theoretical results and the robustness of the numerical scheme.