Mixed-Type Galerkin Variational Principle and Numerical Simulation for a Generalized Nonlocal Elastic Model

Abstract

A mixed-type Galerkin variational principle is proposed for a generalized nonlocal elastic model. The solvability and regularity of its solution is naturally derived through the Lax–Milgram lemma, from which a solvability criterion is inferred for a Fredholm integral equation of the first kind. A mixed-type finite element procedure is therefore developed and the existence and uniqueness of the discrete solution is proved. This compensates the lack of solvability proof for the collocation-finite difference scheme proposed in Du et al. (J Comput Phys 297:72–83, 2015). Numerical error bounds for the unknown and the intermediate variable are proved. By carefully exploring the structure of the coefficient matrices of the numerical method, we develop a fast conjugate gradient algorithm , which reduces the computations to $\mathcal {O}(NlogN)$ per iteration and the memory to $\mathcal {O}(N)$. The use of the preconditioner significantly reduces the number of iterations. Numerical results show the utility of the method.

Appendix

In this appendix, we introduce the definitions and notations of those spaces and the properties of the fractional operators used in this paper. They also can be found in [1, 7, 10, 13, 18, 19] and the references cited therein.

Definition 9.1

(Negative fractional derivative spaces [7, 10]) Let $\mu >0$. Define the norm

$$\begin{aligned} \begin{array}{lll} \Vert v\Vert _{J_L^{-\mu }(\mathbb {R})}:=\left\| _{-\infty }I_x^\mu v\right\| _{L^2(\mathbb {R})}, \end{array} \end{aligned}$$

(9.1)

and let $J_L^{-\mu }(\mathbb {R})$ denote the closure of $C_0^\infty (\mathbb {R})$ with respect to $\Vert \cdot \Vert _{J_L^{-\mu }(\mathbb {R})}$. Analogously, define the norm

$$\begin{aligned} \begin{array}{lll} \Vert v\Vert _{J_R^{-\mu }(\mathbb {R})}:=\left\| _xI_{\infty }^\mu v\right\| _{L^2(\mathbb {R})}, \end{array} \end{aligned}$$

(9.2)

and let $J_R^{-\mu }(\mathbb {R})$ denote the closure of $C_0^\infty (\mathbb {R})$ with respect to $\Vert \cdot \Vert _{J_R^{-\mu }(\mathbb {R})}$.

By the Fourier transforms

$$\begin{aligned} \begin{array}{lll} \mathscr {F}(v)(w)=\int _{-\infty }^\infty e^{-iwx}v(x)dx=\hat{v}(w), \end{array} \end{aligned}$$

the negative fractional order Sobolev space $H^{-\mu }(\mathbb {R})$ is defined to be

Definition 9.2

(Negative Sobolev spaces [1, 18]) Let $\mu >0$. Define the norm

$$\begin{aligned} \Vert v\Vert _{H^{-\mu }(\mathbb {R})}:=\Vert |w|^{-\mu }\hat{v}(w)\Vert _{L^2(\mathbb {R})}, \end{aligned}$$

(9.3)

and let $H^{-\mu }(\mathbb {R})$ denote the closure of $C_0^\infty (\mathbb {R})$ with respect to $\Vert \cdot \Vert _{H^{-\mu }(\mathbb {R})}$.

The following equivalences are proved [10].

Lemma 9.1

([10], Theorem 2.5) The three spaces $J_L^{-\mu }(\mathbb {R})$, $J_R^{-\mu }(\mathbb {R})$ and $H^{-\mu }(\mathbb {R})$ are equal with equivalent norms, and

$$\begin{aligned} \left( _{-\infty } I_x^\mu v,{_x I_\infty ^\mu } v\right) =cos(\mu \pi )\Vert v\Vert ^2_{H^{-\mu }(\mathbb {R})}. \end{aligned}$$

(9.4)

For convenience, we need to restrict the negative fractional derivative spaces to a bounded subinterval of $\mathbb {R}$, which, in this paper, is denoted by (0, 1). Still, we establish their equivalence with $H^{-\mu }(0,1)$.

Definition 9.3

(Negative fractional derivative spaces in bounded domain [7]) Define the spaces $J^{-\mu }_{L}(0,1),J^{-\mu }_{R}(0,1)$ as the closure of $C_0^{\infty }(0,1)$ under their respective norms.

A similar conclusion with Lemma 9.1 can be derived for the bounded domain (0, 1).

Lemma 9.2

([7], Theorem 2.6) Assume that $\mu >0.$ Then, the three spaces $J_L^{-\mu }(0,1)$, $J_R^{-\mu }(0,1)$ and $H^{-\mu }(0,1)$ are equal with equivalent norms, and

$$\begin{aligned} \begin{array}{lll} \left( _0I_x^\mu v,{_xI_1^\mu } v\right) =cos(\mu \pi ) \Vert v\Vert ^2_{H^{-\mu }(0,1)}, \end{array} \end{aligned}$$

(9.5)

We then define the fractional derivative spaces introduced in [13].

Definition 9.4

(Fractional derivative spaces [13]) Let $\mu >0$. Define the semi-norm

$$\begin{aligned} |u|_{J^\mu _L (\mathbb {R})}:=\Vert _{-\infty }D_x^\mu u\Vert _{L^2(\mathbb {R})}, \end{aligned}$$

and norm

$$\begin{aligned} \Vert u\Vert _{J^\mu _L(\mathbb {R})}:=\left( \Vert u\Vert ^2_{L^2(\mathbb {R})}+|u|^2_{J^\mu _L(\mathbb {R})}\right) ^{\frac{1}{2}}, \end{aligned}$$

(9.6)

and let $J^\mu _L(\mathbb {R})$ denote the closure of $C_0^{\infty }(\mathbb {R})$ with respect to $\Vert \cdot \Vert _{J^\mu _L(\mathbb {R})}$.

Analogously, we define the right fractional derivative space as follows. Let $\mu >0$. Define the semi-norm

$$\begin{aligned} |u|_{J^\mu _R (\mathbb {R})}:=\Vert _xD_{+\infty }^\mu u\Vert _{L^2(\mathbb {R})}, \end{aligned}$$

and norm

$$\begin{aligned} \Vert u\Vert _{J^\mu _R(\mathbb {R})}:=\left( \Vert u\Vert ^2_{L^2(\mathbb {R})}+|u|^2_{J^\mu _R(\mathbb {R})}\right) ^{\frac{1}{2}}, \end{aligned}$$

(9.7)

and let $J^\mu _R(\mathbb {R})$ denote the closure of $C_0^{\infty }(\mathbb {R})$ with respect to $\Vert \cdot \Vert _{J^\mu _R(\mathbb {R})}$.

We also define the norm for functions in $H^\mu (\mathbb {R})$ in terms of the Fourier transforms.

Definition 9.5

(Fractional Sobolev spaces [1, 13, 18]) Let $\mu >0$. Define the semi-norm

$$\begin{aligned} |u|_{H^\mu (\mathbb {R})}:=\Vert |w|^\mu \hat{u}\Vert _{L^2(\mathbb {R})}, \end{aligned}$$

and norm

$$\begin{aligned} \Vert u\Vert _{H^\mu (\mathbb {R})}:=\left( \Vert u\Vert ^2_{L^2(\mathbb {R})}+|u|^2_{H^\mu (\mathbb {R})}\right) ^{1/2}, \end{aligned}$$

(9.8)

and let $H^\mu (\mathbb {R})$ denotes the closure of $C_0^\infty (\mathbb {R})$ with respect to $\Vert \cdot \Vert _{H^\mu (\mathbb {R})}$.

As stated in [13], the product of the left and the right fractional order derivative for the same real valued function u can be related to $|\cdot |_{J_L^\mu (\mathbb {R})}$.

Lemma 9.3

([13], Lemma 2.4) Assume that $\mu >0$. Then for u(x) a real valued function

$$\begin{aligned} \left( _{-\infty }D_x^\mu u,{_xD_{+\infty }^\mu } u\right) =cos( \mu \pi )\Vert _{-\infty }D_x^{\mu }u\Vert ^2_{L^2(\mathbb {R})}. \end{aligned}$$

(9.9)

At last, several good conclusions of the fractional operators will be demonstrated using the following lemmas.

Lemma 9.4

(Semigroup property for fractional integrate operator [19]) Let $\mu ,\nu >0.$ For any $u \in L^2(0,1),$ we have

$$\begin{aligned} _0I_x^{\mu +\nu }u={_0I_x^\mu }{_0I_x^\nu }u. \end{aligned}$$

(9.10)

Lemma 9.5

(Adjoint property for fractional integrate operator [19]) Let $\mu >0.$ For any $u,v \in L^2(0,1),$ we have

$$\begin{aligned} (_0I_x^\mu u,v)=(u, {_xI_1^\mu }v). \end{aligned}$$

(9.11)

Lemma 9.6

(Semigroup property for fractional derivative operator [13]) Let $0<s<\mu .$ For any $u \in J^\mu _{L,0}(0,1)$, we have

$$\begin{aligned} _{0}D_x^\mu u={_0D_x^s}{_0D_x^{\mu -s}}u, \end{aligned}$$

(9.12)

and similarly for any $u\in J^\mu _{R,0}(0,1)$, we have

$$\begin{aligned} _xD_{1}^\mu u={_xD_{1}^s}{_xD_{1}^{\mu -s}}u. \end{aligned}$$

(9.13)

Lemma 9.7

(Adjoint property for fractional derivative operator [13]) Let $1<\beta <2$. For any $\omega \in H^\beta _0(0,1),v \in H_0^{\frac{\beta }{2}}(0,1)$, we have

$$\begin{aligned} \left( _0D_x^\beta \omega ,v\right) _{(0,1)}=\left( _0D_x^{\frac{\beta }{2}}\omega ,{_xD_1^{\frac{\beta }{2}}} v\right) _{(0,1)}, \end{aligned}$$

(9.14)

$$\begin{aligned} \left( _xD_1^\beta \omega ,v\right) _{(0,1)}=\left( _xD_1^{\frac{\beta }{2}}\omega ,{_0D_x^{\frac{\beta }{2}}} v\right) _{(0,1)}. \end{aligned}$$

(9.15)