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

TL;DR

This paper introduces a novel Petrov-Galerkin finite element method for one-dimensional fractional boundary value problems involving Riemann-Liouville or Caputo derivatives, providing optimal error estimates and efficient linear systems.

Contribution

It develops a new variational formulation and finite element scheme with shifted fractional powers, enabling optimal error bounds and well-conditioned matrices for fractional convection-diffusion equations.

Findings

Optimal error estimates in L2 and H1 norms.

Diagonal stiffness matrix on uniform meshes.

Enriched FEM improves convergence in Riemann-Liouville case.

Abstract

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 $\alpha\in(3/2, 2)$ in the leading term and both convection and potential terms. They arise in the mathematical modeling of asymmetric super-diffusion 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 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 deriving optimal error estimates in both $L^2(D)$ and $H^1(D)$ 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.