A Petrov--Galerkin Finite Element Method for Fractional Convection-Diffusion Equations Academic Article uri icon

abstract

  • 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 (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.

published proceedings

  • SIAM Journal on Numerical Analysis

author list (cited authors)

  • Jin, B., Lazarov, R., & Zhou, Z.

citation count

  • 42

complete list of authors

  • Jin, Bangti||Lazarov, Raytcho||Zhou, Zhi

publication date

  • January 2016