A Petrov--Galerkin Finite Element Method for Fractional Convection-Diffusion Equations
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© 2016 Society for Industrial and Applied Mathematics. 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.
author list (cited authors)
Jin, B., Lazarov, R., & Zhou, Z.