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

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          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.

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          Journal
          17 December 2015
          Article
          1512.05634
          5ab82781-d3d9-48e9-8564-9013e6c9c8e2

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          23 pp
          math.NA

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