# An enriched finite element method to fractional advection–diffusion equation

An enriched finite element method to fractional advection–diffusion equation In this paper, an enriched finite element method with fractional basis $$\left[ 1,x^{\alpha }\right]$$ 1 , x α 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 $$\left[ 1,x\right]$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Mechanics Springer Journals

# An enriched finite element method to fractional advection–diffusion equation

, Volume 60 (2) – Mar 21, 2017
21 pages

Publisher
Springer Berlin Heidelberg
Subject
Engineering; Theoretical and Applied Mechanics; Computational Science and Engineering; Classical and Continuum Physics
ISSN
0178-7675
eISSN
1432-0924
D.O.I.
10.1007/s00466-017-1400-9
Publisher site
See Article on Publisher Site

### Abstract

In this paper, an enriched finite element method with fractional basis $$\left[ 1,x^{\alpha }\right]$$ 1 , x α 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 $$\left[ 1,x\right]$$ 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.

### Journal

Computational MechanicsSpringer Journals

Published: Mar 21, 2017

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