A Petrov--Galerkin Finite Element Method for Fractional Convection-Diffusion Equations

In this work, we develop variational formulations of Petrov--Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann--Liouville or Caputo derivative of order α∈(3/2, 2) in the leading term and both convection and potential terms. They arise in the mathematical modeling of asymmetric superdiffusion processes in heterogeneous media. The well-posedness of the formulations and sharp regularity pickup of the variational solutions are established. A novel finite element method (FEM) is developed, which employs continuous piecewise linear finite elements and “shifted” fractional powers for the trial and test space, respectively. The new approach has a number of distinct features: it allows the derivation of optimal error estimates in both the L²(D) and H¹(D) norms; and on a uniform mesh, the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned. Further, in the Riemann--Liouville case, an enriched FEM is proposed to improve the convergence. Extensive numerical results are presented to verify the theoretical analysis and robustness of the numerical scheme.