A PETROV-GALERKIN FINITE ELEMENT METHOD FOR FRACTIONAL CONVECTION-DIFFUSION EQUATIONS

作者：Jin Bangti; Lazarov Raytcho; Zhou Zhi

来源：SIAM Journal on Numerical Analysis, 2016, 54(1): 481-503.

DOI：10.1137/140992278

• fractional convection-diffusion equation
• variational formulation
• finite element method
• optimal error estimates

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 is an element of (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.

• 出版日期2016