An enriched finite element method to fractional advection–diffusion equation

Shengzhi Luan, Yanping Lian*, Yuping Ying, Shaoqiang Tang, Gregory J. Wagner, Wing Kam Liu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


In this paper, an enriched finite element method with fractional basis [1 , xα] for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis [1 , x]. For fractional advection–diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov–Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection–diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases.

Original languageEnglish (US)
Pages (from-to)181-201
Number of pages21
JournalComputational Mechanics
Issue number2
StatePublished - Aug 1 2017


  • Anomalous diffusion
  • Fractional advection–diffusion equation
  • Fractional calculus
  • Petrov–Galerkin formulation

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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