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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Jin, Bangti Lazarov, Raytcho Zhou, Zhi
Copyright Year	2016
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 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 $L^2(D)$ 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.
Starting Page	481
Ending Page	503
Page Count	23
File Format	PDF
ISSN	00361429
DOI	10.1137/140992278
e-ISSN	10957170
Issue Number	1
Volume Number	54
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-02-25
Access Restriction	Subscribed
Subject Keyword	variational formulation finite element method Error bounds optimal error estimates Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods fractional convection-diffusion equation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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