An enriched finite element method to fractional advection–diffusion equation

Abstract

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

Appendices

Appendix 1

In this appendix, the following formula is demonstrated.

$$\begin{aligned}&\sum _{J=1}^{k,k\le I-2}\check{K}_{IJ}\left( \alpha ,M\right) \phi _{J}^{\text {ex}} =\frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{k^{\alpha }\left( k+1\right) ^{\alpha }}{\left( k+1\right) ^{\alpha }-k^{\alpha }}\nonumber \\&\quad \times \frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}\left( M_{G}^{\alpha }\mid _{\frac{k}{I-1}}^{\frac{k}{I}}-M_{G}^{\alpha }\mid _{\frac{k+1}{I-1}}^{\frac{k+1}{I}}\right) \nonumber \\&\qquad - \frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{k^{\alpha }\left( k+1\right) ^{\alpha }}{\left( k+1\right) ^{\alpha }-k^{\alpha }}\nonumber \\&\quad \times \frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}\left( M_{G}^{\alpha }\mid _{\frac{k}{I}}^{\frac{k}{I+1}}-M_{G}^{\alpha }\mid _{\frac{k+1}{I}}^{\frac{k+1}{I+1}}\right) , \end{aligned}$$

(72)

where $\check{K}_{IJ}\left( \alpha ,M\right)$ is part of stiffness matrix following the formula in Eq. (17), and $\phi _{J}^{\text {ex}}\left( x\right) =\left( J\triangle x\right) ^{\alpha }$ is the nodal exact solution. Hence, Eq. (72)can be rewritten as

$$\begin{aligned} \sum _{J=1}^{k,k\le I-2}\check{K}_{IJ}\left( \alpha ,M\right) \phi _{J}^{\text {ex}}=\frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\sum _{J=1}^{k,k\le I-2}A_{J}B_{J}. \end{aligned}$$

(73)

The mathematical expression of $A_{J}$ and $B_{J}$ in Eq. (73) are defined as

$$\begin{aligned} A_{J}= & {} \frac{J^{\alpha }}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] \left[ J^{\alpha }-\left( J-1\right) ^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J}{I-1}}^{\frac{J}{I}}\nonumber \\&-\,\frac{\left( J-1\right) ^{\alpha }}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] \left[ J^{\alpha }-\left( J-1\right) ^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J-1}{I-1}}^{\frac{J-1}{I}}\nonumber \\&+\,\frac{J^{\alpha }}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] \left[ \left( J+1\right) ^{\alpha }-J^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J}{I-1}}^{\frac{J}{I}}\nonumber \\&-\,\frac{\left( J+1\right) ^{\alpha }}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] \left[ \left( J+1\right) ^{\alpha }-J^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J+1}{I-1}}^{\frac{J+1}{I}}\nonumber \\&-\,\frac{J^{\alpha }}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] \left[ J^{\alpha }-\left( J-1\right) ^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J}{I}}^{\frac{J}{I+1}}\nonumber \\&+\,\frac{\left( J-1\right) ^{\alpha }}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] \left[ J^{\alpha }-\left( J-1\right) ^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J-1}{I}}^{\frac{J-1}{I+1}}\nonumber \\&-\,\frac{J^{\alpha }}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] \left[ \left( J+1\right) ^{\alpha }-J^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J}{I}}^{\frac{J}{I+1}}\nonumber \\&+\,\frac{\left( J+1\right) ^{\alpha }}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] \left[ \left( J+1\right) ^{\alpha }-J^{\alpha }\right] }M_{G}^{\alpha }\mid _{\frac{J+1}{I}}^{\frac{J+1}{I+1}}. \end{aligned}$$

(74)

$$\begin{aligned} B_{J}= & {} J^{\alpha }. \end{aligned}$$

(75)

By mathematical induction, Eq. (72) can be demonstrated as following steps.

$$\begin{aligned} S_{1}= & {} A_{1}B_{1}=\frac{1^{\alpha }2^{\alpha }}{2^{\alpha }-1^{\alpha }}\left[ \frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{1}{I-1}}^{\frac{1}{I}}\right. \nonumber \\&-\,\frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{1}{I}}^{\frac{1}{I+1}}+\frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{2}{I}}^{\frac{2}{I+1}}\nonumber \\&\left. -\,\frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{2}{I-1}}^{\frac{2}{I}}\right] . \end{aligned}$$

(76)

Assuming

$$\begin{aligned} S_{n}=&\sum _{J=1}^{n}A_{J}B_{J}=\frac{n^{\alpha }\left( n+1\right) ^{\alpha }}{\left( n+1\right) ^{\alpha }-n^{\alpha }}\left[ \frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n}{I-1}}^{\frac{n}{I}}\right. \nonumber \\&\left. -\,\frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n}{I}}^{\frac{n}{I+1}}\right] \nonumber \\&+\, \frac{n^{\alpha }\left( n+1\right) ^{\alpha }}{\left( n+1\right) ^{\alpha }-n^{\alpha }}\left[ \frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+1}{I}}^{\frac{n+1}{I+1}}\right. \nonumber \\&\left. -\,\frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+1}{I-1}}^{\frac{n+1}{I}}\right] . \end{aligned}$$

(77)

Then, by some simple algerbaric calculations

$$\begin{aligned} S_{n+1}&=\sum _{J=1}^{n+1}A_{J}B_{J}=\frac{\left( n+1\right) ^{\alpha }\left( n+2\right) ^{\alpha }}{\left( n+2\right) ^{\alpha }-\left( n+1\right) ^{\alpha }}\nonumber \\&\times \left[ \frac{1}{I^{\alpha }{-}\left( I{-}1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+1}{I-1}}^{\frac{n+1}{I}}{-}\frac{1}{\left( I{+}1\right) ^{\alpha }{-}I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+1}{I}}^{\frac{n+1}{I+1}}\right] \nonumber \\&\quad +\, \frac{\left( n+1\right) ^{\alpha }\left( n+2\right) ^{\alpha }}{\left( n+2\right) ^{\alpha }-\left( n+1\right) ^{\alpha }}\left[ \frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+2}{I}}^{\frac{n+2}{I+1}}\right. \nonumber \\&\quad \left. -\,\frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}M_{G}^{\alpha }\mid _{\frac{n+2}{I-1}}^{\frac{n+2}{I}}\right] . \end{aligned}$$

(78)

Hence, Eq. (72) is proved.

$$\begin{aligned}&\sum _{J=1}^{k,k\le I-2}\check{K}_{IJ}\left( \alpha ,M\right) \phi _{J}^{\text {ex}} =\frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }S_{k}\nonumber \\&\quad = \frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{k^{\alpha }\left( k+1\right) ^{\alpha }}{\left( k+1\right) ^{\alpha }-k^{\alpha }}\nonumber \\&\qquad \times \frac{1}{I^{\alpha }-\left( I-1\right) ^{\alpha }}\left( M_{G}^{\alpha }\mid _{\frac{k}{I-1}}^{\frac{k}{I}}-M_{G}^{\alpha }\mid _{\frac{k+1}{I-1}}^{\frac{k+1}{I}}\right) \nonumber \\&\qquad - \frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{k^{\alpha }\left( k+1\right) ^{\alpha }}{\left( k+1\right) ^{\alpha }-k^{\alpha }}\nonumber \\&\qquad \times \frac{1}{\left( I+1\right) ^{\alpha }-I^{\alpha }}\left( M_{G}^{\alpha }\mid _{\frac{k}{I}}^{\frac{k}{I+1}}-M_{G}^{\alpha }\mid _{\frac{k+1}{I}}^{\frac{k+1}{I+1}}\right) . \end{aligned}$$

(79)

Appendix 2

In this appendix, the appropriate fractional element Peclet number $\hat{P}e$ is formulated, and the matrix form of Galekrin discretization is rewritten with fractional element Peclet number.

Based on the formula of fractional diffusion stiffness matrix in Eq. (15), $\varvec{K}\left( \alpha ,M\right)$ can be rewritten as

$$\begin{aligned}&\varvec{K}\left( \alpha ,M\right) =\left[ \begin{array}{ccccccc} \lambda _{1} &{} \lambda _{2} &{} \lambda _{3} &{} \cdots &{} \cdots &{} \cdots &{} \lambda _{M-1}\\ \lambda _{1} &{} \lambda _{2} &{} \lambda _{3} &{} \cdots &{} \cdots &{} \cdots &{} \lambda _{M-1}\\ \lambda _{1} &{} \lambda _{2} &{} \lambda _{3} &{} \cdots &{} \cdots &{} \cdots &{} \lambda _{M-1}\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \lambda _{1} &{} \lambda _{2} &{} \lambda _{3} &{} \cdots &{} \cdots &{} \cdots &{} \lambda _{M-1} \end{array}\right] \nonumber \\&\quad \times \left[ \begin{array}{ccccccc} \omega _{11} &{} -1\\ \omega _{22} &{} \omega _{21} &{} -1 &{} &{} &{} 0\\ \omega _{33} &{} \omega _{32} &{} \omega _{31} &{} -1\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} &{} -1\\ \omega _{M-1,1} &{} \omega _{M-1,2} &{} \omega _{M-1,3} &{} \cdots &{} \cdots &{} \cdots &{} \omega _{M-1,M-1} \end{array}\right] .\nonumber \\ \end{aligned}$$

(80)

The stiffness matrix $\varvec{K}\left( 0,M\right)$ is calculated as

$$\begin{aligned} K_{IJ}\left( 0,M\right) =\left\{ \begin{array}{lll} -0.5, &{} &{} J=I+1\\ 0, &{} &{} J=I\\ 0.5, &{} &{} J=I-1. \end{array}\right. \end{aligned}$$

(81)

Therefore, the matrix form of fractional advection–diffusion equation can be rewritten as

$$\begin{aligned} -\tilde{\varvec{K}}\left( 0,M\right) +\tilde{\varvec{K}}\left( \alpha ,M\right) =\tilde{\varvec{F}}, \end{aligned}$$

(82)

where,

$$\begin{aligned}&\tilde{\varvec{K}}\left( 0,M\right) =\left[ \begin{array}{lllllll} 0 &{} -1\\ 1 &{} 0 &{} -1\\ &{} 1 &{} 0 &{} -1\\ &{} &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} &{} &{} \ddots &{} \ddots &{} -1\\ &{} &{} &{} &{} &{} 1 &{} 0 \end{array}\right] ,\end{aligned}$$

(83)

$$\begin{aligned}&\tilde{\varvec{K}}\left( \alpha ,M\right) =\left[ \begin{array}{ccccccc} \bar{\lambda }_{1} &{} \bar{\lambda }_{2} &{} \bar{\lambda }_{3} &{} \cdots &{} \cdots &{} \cdots &{} \bar{\lambda }_{M-1}\\ \bar{\lambda }_{1} &{} \bar{\lambda }_{2} &{} \bar{\lambda }_{3} &{} \cdots &{} \cdots &{} \cdots &{} \bar{\lambda }_{M-1}\\ \bar{\lambda }_{1} &{} \bar{\lambda }_{2} &{} \bar{\lambda }_{3} &{} \cdots &{} \cdots &{} \cdots &{} \bar{\lambda }_{M-1}\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \bar{\lambda }_{1} &{} \bar{\lambda }_{2} &{} \bar{\lambda }_{3} &{} \cdots &{} \cdots &{} \cdots &{} \bar{\lambda }_{M-1} \end{array}\right] \nonumber \\&\quad \times \left[ \begin{array}{ccccccc} \omega _{11} &{} -1\\ \omega _{22} &{} \omega _{21} &{} -1 &{} &{} &{} 0\\ \omega _{33} &{} \omega _{32} &{} \omega _{31} &{} -1\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} &{} -1\\ \omega _{M-1,1} &{} \omega _{M-1,2} &{} \omega _{M-1,3} &{} \cdots &{} \cdots &{} \cdots &{} \omega _{M-1,M-1} \end{array}\right] ,\nonumber \\ \end{aligned}$$

(84)

$$\begin{aligned}&\tilde{\varvec{F}}=\left[ \begin{array}{c} 0\\ 0\\ 0\\ \vdots \\ \vdots \\ 0\\ \bar{\lambda }_{M}-1 \end{array}\right] , \end{aligned}$$

(85)

where

$$\begin{aligned} \bar{\lambda }_{I}= & {} \frac{2v}{u\triangle x^{\alpha }}\left[ \frac{\varGamma \left( 1+\alpha \right) }{\left( I+1\right) ^{\alpha }-I^{\alpha }}-\frac{\alpha \varGamma \left( 1+\alpha \right) }{\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\right. \nonumber \\&\quad \times \left. \frac{I^{\alpha }}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] ^{2}}M_{G}^{\alpha }\mid _{1}^{\frac{I}{I+1}}\right] =\frac{2v}{u\triangle x^{\alpha }}\vartheta _{I}. \end{aligned}$$

(86)

Here, $\vartheta _{I}$ always has almost the same value of $\frac{1}{\varGamma \left( 3-\alpha \right) }$, which can be numerically demonstrated as shown in Fig. 12.

Fig. 12

Comparison between $\vartheta _{I}$ and $\frac{1}{\varGamma \left( 3-\alpha \right) }$ for different fractional order a $\alpha =0.3$, b $\alpha =0.5$ and c $\alpha =0.7$

Hence, we assume all the element value of $\vartheta _{I}$ has the same value of $\frac{1}{\varGamma \left( 3-\alpha \right) }$ and then all the element value of $\bar{\lambda }_{I}$ becomes $\frac{2v}{u\triangle x^{\alpha }\varGamma \left( 3-\alpha \right) }$.

Therefore, the matrix form in Eq. (82) can be rewritten as

$$\begin{aligned}&-\hat{P}e\left[ \begin{array}{lllllll} 0 &{} -1\\ 1 &{} 0 &{} -1\\ &{} 1 &{} 0 &{} -1\\ &{} &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} &{} &{} \ddots &{} \ddots &{} -1\\ &{} &{} &{} &{} &{} 1 &{} 0 \end{array}\right] \nonumber \\&\qquad +\left[ \begin{array}{ccccccc} \omega _{11} &{} -1\\ \omega _{22} &{} \omega _{21} &{} -1 &{} &{} &{} 0\\ \omega _{33} &{} \omega _{32} &{} \omega _{31} &{} -1\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} \ddots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} &{} &{} -1\\ \omega _{M-1,1} &{} \omega _{M-1,2} &{} \omega _{M-1,3} &{} \cdots &{} \cdots &{} \cdots &{} \omega _{M-1,M-1} \end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{c} 0\\ 0\\ 0\\ \vdots \\ \vdots \\ 0\\ 1-\hat{P}e \end{array}\right] . \end{aligned}$$

(87)

The matrix form of Eq. (87) is denoted as

$$\begin{aligned} -\hat{P}e\bar{\varvec{K}}\left( 0,M\right) +\bar{\varvec{K}}\left( \alpha ,M\right) =\bar{F}. \end{aligned}$$

(88)

Appendix 3

In this appendix, the detailed formula of stiffness matrices $\varvec{K}\left( 1,M\right)$ and $\varvec{K}\left( 1+\alpha ,M\right)$ are derived as follows.

$\varvec{K}\left( 1,M\right)$ is a tri-diagonal matrix, formulated as

$$\begin{aligned}&K_{IJ}\left( 1,M\right) =\frac{\alpha ^{2}}{2\alpha -1}\frac{1}{\triangle x}\nonumber \\&\quad \times \left\{ \begin{array}{lll} -\frac{J^{2\alpha -1}-I^{2\alpha -1}}{\left( J^{\alpha }-I^{\alpha }\right) ^{2}}, &{} &{} J=I+1\\ \frac{\left( I+1\right) ^{2\alpha -1}-I^{2\alpha -1}}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] ^{2}}+\frac{I^{2\alpha -1}-\left( I-1\right) ^{2\alpha -1}}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] ^{2}}, &{} &{} J=I\\ -\frac{I^{2\alpha -1}-J^{2\alpha -1}}{\left( I^{\alpha }-J^{\alpha }\right) ^{2}}, &{} &{} J=I-1. \end{array}\right. \end{aligned}$$

(89)

$\varvec{K}\left( 1+\alpha ,M\right)$ is a lower triangular matrix, formulated as

$$\begin{aligned}&K_{IJ}\left( 1+\alpha ,M\right) =\frac{1}{\varGamma \left( 2-\alpha \right) }\frac{\alpha ^{2}}{\triangle x^{1+\alpha }}\nonumber \\&\quad \times \left\{ \begin{array}{lll} 0, &{} &{} J\ge I+2\\ -\frac{1}{\left( J^{\alpha }-I^{\alpha }\right) ^{2}}\frac{1}{\left( IJ\right) ^{1-\alpha }}, &{} &{} J=I+1\\ \frac{1}{\left[ \left( I+1\right) I\right] ^{1-\alpha }}\left\{ \frac{1}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] \left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] }+\frac{1}{\left[ \left( I+1\right) ^{\alpha }-I^{\alpha }\right] ^{2}}\right\} , &{} &{} J=I\\ \sum _{m=I-1}^{I}\sum _{n=J-1}^{J}\frac{1}{\left( m+1\right) ^{\alpha }-m^{\alpha }}\frac{1}{\left( n+1\right) ^{\alpha }-n^{\alpha }}\\ \left[ \sum _{p=m}^{m+1}\sum _{q=n}^{n+1}\left( -1\right) ^{p+q-I-J+1}\left( \frac{p-q}{pq}\right) ^{1-\alpha }\right] -\frac{1}{\left[ I^{\alpha }-\left( I-1\right) ^{\alpha }\right] ^{2}}\frac{1}{I^{1-\alpha }\left( I-1\right) ^{1-\alpha }}, &{} &{} J=I-1\\ \sum _{m=I-1}^{I}\sum _{n=J-1}^{J}\frac{1}{\left( m+1\right) ^{\alpha }-m^{\alpha }}\frac{1}{\left( n+1\right) ^{\alpha }-n^{\alpha }}\\ \left[ \sum _{p=m}^{m+1}\sum _{q=n}^{n+1}\left( -1\right) ^{p+q-I-J+1}\left( \frac{p-q}{pq}\right) ^{1-\alpha }\right] , &{} &{} J\le I-2. \end{array}\right. \end{aligned}$$

(90)

From the formula of stiffness $\varvec{K}\left( 1,M\right)$ , one special case is that $K_{11}\left( 1,M\right)$ may not exist when the fractional order $\alpha <0.5$, which reads

$$\begin{aligned} K_{11}\left( 1,M\right) =\frac{\alpha ^{2}}{\triangle x}\left[ \int _{0}^{\triangle x}x^{2\alpha -2}\text {d}x+\int _{\triangle x}^{2\triangle x}x^{2\alpha -2}\text {d}x\right] . \end{aligned}$$

(91)

Therefore, we rewritte the formula of $K_{11}\left( 1,M\right)$ as

$$\begin{aligned}&K_{11}\left( 1,M\right) \!=\!\frac{\alpha ^{2}}{\triangle x\left( 2\alpha -1\right) \left( 2^{\alpha }-1\right) ^{2}}\nonumber \\&\quad \times \left\{ \begin{array}{lll} \left( 2^{2\alpha -1}\!-\!1\right) +2^{1-2\alpha }\left( 2\alpha -1\right) \left( 2^{\alpha }\!-\!1\right) ^{2}\varepsilon , &{} &{} \alpha \le 0.5\\ \left( 2^{2\alpha -1}-1\right) +\left( 2^{\alpha }-1\right) ^{2}, &{} &{} \alpha >0.5, \end{array}\right. \nonumber \\ \end{aligned}$$

(92)

where $\varepsilon$ represents the approximation of integration $\left( \frac{2}{\triangle x}\right) ^{2\alpha -1}\int _{0}^{\triangle x}x^{2\alpha -2}\text {d}x$. In this article, it is recommended by the author to use Gaussian quadrature approximation to calculate $\varepsilon$ in Eq. (92), and takes three Gaussian points as an example, $\varepsilon$ is calculated as

$$\begin{aligned}&\varepsilon =\left( \frac{2}{\triangle x}\right) ^{2\alpha -1}\int _{0}^{\triangle x}x^{2\alpha -2}\text {d}x=\int _{-1}^{1}\left( 1+\xi \right) ^{2\alpha -2}\text {d}\xi \nonumber \\&\quad \thickapprox \frac{8}{9}+\frac{5}{9}\left[ \left( 1-\sqrt{0.6}\right) ^{2\alpha -2}+\left( 1+\sqrt{0.6}\right) ^{2\alpha -2}\right] . \end{aligned}$$

(93)

Additionally, the load vector $\varvec{F}$ is formulated as

$$\begin{aligned} F_{I}=\left\{ \begin{array}{ll@{\quad }l} 0, &{} &{} I<M-1\\ -\frac{1}{2}u+v\left\{ \frac{\varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }}\frac{1}{M^{\alpha }-\left( M-1\right) ^{\alpha }}-\frac{\alpha \varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{\left( M-1\right) ^{\alpha }}{\left[ M^{\alpha }-\left( M-1\right) ^{\alpha }\right] ^{2}}M_{G}^{\alpha }\mid _{1}^{\frac{M-1}{M}}\right\} \\ +u\gamma \frac{\alpha ^{2}}{2\alpha -1}\frac{1}{\triangle x}\frac{M^{2\alpha -1}-\left( M-1\right) ^{2\alpha -1}}{\left[ M^{\alpha }-\left( M-1\right) ^{\alpha }\right] ^{2}}-v\gamma \frac{1}{\varGamma \left( 2-\alpha \right) }\frac{\alpha ^{2}}{\triangle x^{1+\alpha }}\frac{1}{\left[ M^{\alpha }-\left( M-1\right) ^{\alpha }\right] ^{2}}\frac{1}{\left[ M\left( M-1\right) \right] ^{1-\alpha }}, &{} &{} I=M-1. \end{array}\right. \end{aligned}$$

(94)

Appendix 4

In this appendix, the detailed derivations of the artificial viscosity coefficient of the enriched Petrov–Galerkin finite element method to fractional advection–diffusion equation is proposed.

The load vector residual is defined as

$$\begin{aligned} \varvec{R}=\left\| \varvec{F}^{*}-\varvec{F}\right\| , \end{aligned}$$

(95)

Considering the first term in Eq. (95), the load vector at first node reads

$$\begin{aligned} F_{1}= & {} 0,\end{aligned}$$

(96)

$$\begin{aligned} F_{1}^{*}= & {} \left[ -uK_{11}\left( 0,M\right) +vK_{11}\left( \alpha ,M\right) \right. \nonumber \\&\left. +u\gamma K_{11}\left( 1,M\right) -v\gamma K_{11}\left( 1+\alpha ,M\right) \right] \phi _{1}^{\text {ex}}\nonumber \\&+ \left[ -uK_{12}\left( 0,M\right) +vK_{12}\left( \alpha ,M\right) +u\gamma K_{12}\left( 1,M\right) \right. \nonumber \\&\left. -v\gamma K_{12}\left( 1+\alpha ,M\right) \right] \phi _{2}^{\text {ex}} \end{aligned}$$

(97)

Based on the formulas of stiffness matrix in Eq. (15), (81), (89), (90), and exact nodal solution in Eq. (37), the details in Eq. (97) are formulated as

$$\begin{aligned}&K_{11}\left( 0,M\right) =0,\end{aligned}$$

(98)

$$\begin{aligned}&K_{12}\left( 0,M\right) =-\frac{1}{2},\end{aligned}$$

(99)

$$\begin{aligned}&K_{11}\left( \alpha ,M\right) =\frac{\varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }}\frac{2^{\alpha }}{2^{\alpha }-1}\nonumber \\&\quad -\,\frac{\alpha \varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{2^{\alpha }}{\left( 2^{\alpha }-1\right) ^{2}}M_{G}^{\alpha }\mid _{1}^{0.5},\end{aligned}$$

(100)

$$\begin{aligned}&K_{12}\left( \alpha ,M\right) =-\frac{\varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }}\frac{1}{2^{\alpha }-1}\nonumber \\&\quad +\,\frac{\alpha \varGamma \left( 1+\alpha \right) }{\triangle x^{\alpha }\varGamma \left( 1-\alpha \right) \varGamma ^{2}\left( \alpha \right) }\frac{1}{\left( 2^{\alpha }-1\right) ^{2}}M_{G}^{\alpha }\mid _{1}^{0.5}, \end{aligned}$$

(101)

$$\begin{aligned}&K_{11}\left( 1,M\right) =\frac{\alpha ^{2}}{\triangle x\left( 2\alpha -1\right) \left( 2^{\alpha }-1\right) ^{2}}\nonumber \\&\quad \times \left\{ \begin{array}{lll} \left( 2^{2\alpha {-}1}{-}1\right) +2^{1-2\alpha }\left( 2\alpha -1\right) \left( 2^{\alpha }{-}1\right) ^{2}\varepsilon , &{} &{} \alpha \le 0.5\\ \left( 2^{2\alpha {-}1}{-}1\right) +\left( 2^{\alpha }{-}1\right) ^{2}, &{} &{} \alpha >0.5, \end{array}\right. \nonumber \\ \end{aligned}$$

(102)

$$\begin{aligned}&K_{12}\left( 1,M\right) =-\frac{2^{2\alpha -1}-1}{\left( 2^{\alpha }-1\right) ^{2}}\frac{\alpha ^{2}}{2\alpha -1}\frac{1}{\triangle x}, \end{aligned}$$

(103)

$$\begin{aligned}&K_{11}\left( 1+\alpha ,M\right) =\frac{1}{\varGamma \left( 2-\alpha \right) }\frac{2^{2\alpha -1}}{\left( 2^{\alpha }-1\right) ^{2}}\frac{\alpha ^{2}}{\triangle x^{1+\alpha }}, \end{aligned}$$

(104)

$$\begin{aligned}&K_{12}\left( 1+\alpha ,M\right) =-\frac{1}{\varGamma \left( 2-\alpha \right) }\frac{2^{\alpha -1}}{\left( 2^{\alpha }-1\right) ^{2}}\frac{\alpha ^{2}}{\triangle x^{1+\alpha }}, \end{aligned}$$

(105)

$$\begin{aligned}&\phi _{1}^{\text {ex}}=\frac{E_{\alpha ,1}\left( \frac{u}{v}\triangle x^{\alpha }\right) -1}{E_{\alpha ,1}\left( \frac{u}{v}\right) -1}, \end{aligned}$$

(106)

$$\begin{aligned}&\phi _{2}^{\text {ex}}=\frac{E_{\alpha ,1}\left( \frac{u}{v}\left( 2\triangle x\right) ^{\alpha }\right) -1}{E_{\alpha ,1}\left( \frac{u}{v}\right) -1}. \end{aligned}$$

(107)

Substituting all the formulas above into Eq. (97), by expanding and minimizing $R_{1}=\left| F_{1}^{*}-F_{1}\right| =0$, the artificial viscosity coefficient is formulated as

$$\begin{aligned} \gamma =\frac{\triangle x}{2}\frac{\hat{P}e\left( 2^{\alpha }-1\right) ^{2}\varTheta +\alpha \varGamma \left( 3-\alpha \right) \left( 2^{\alpha }-\varTheta \right) \left[ \left( 2^{\alpha }-1\right) \varGamma \left( \alpha \right) -\frac{\alpha sin\left( \pi \alpha \right) }{\pi }M_{G}^{\alpha }\mid _{1}^{0.5}\right] }{\left[ 2^{\alpha -2}\left( 2^{\alpha }-\varTheta \right) \left( 2-\alpha \right) -\hat{P}e\chi \right] \alpha ^{2}}, \end{aligned}$$

(108)

where $\varTheta$ is a function of fractional Peclet number defined as

$$\begin{aligned} \varTheta =\frac{E_{\alpha ,1}\left( 2^{1+\alpha }\hat{P}e/\varGamma \left( 3-\alpha \right) \right) -1}{E_{\alpha ,1}\left( 2\hat{P}e/\varGamma \left( 3-\alpha \right) \right) -1}. \end{aligned}$$

(109)

$\chi$ is a function of parameter $\varepsilon$ introduced by the special case of $K_{11}\left( 1,M\right)$ as shown in “Appendix 3”, and defined as

$$\begin{aligned} \chi =\frac{1}{2\alpha -1}\left\{ \begin{array}{lll} \left( 2^{2\alpha -1}-1\right) \left( 1-\varTheta \right) \\ \quad +2^{1-2\alpha }\left( 2\alpha -1\right) \left( 2^{\alpha }-1\right) ^{2}\varepsilon , &{} &{} \alpha \le 0.5\\ \left( 2^{2\alpha -1}-1\right) \left( 1-\varTheta \right) \\ \quad +\left( 2^{\alpha }-1\right) ^{2}, &{} &{} \alpha >0.5. \end{array}\right. \end{aligned}$$

(110)