This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in \(\mathbb{R}^{n}\) (\(n\geq2\)). We combine the Morley element method with the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to propose multigrid discretization schemes in an adaptive fashion. We establish an inequality on Rayleigh quotient and use it to prove the efficiency of the adaptive algorithms. Numerical experiments show that these algorithms are efficient and can get the optimal convergence rate.
Notes
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1 Introduction
Biharmonic equation/eigenvalue problem plays an important role in elastic mechanics. In 1968, Morley designed a famous non-conforming element called the Morley element [1] to solve biharmonic equation (plate bending problem). The Morley element was extended to arbitrarily dimensions by Wang and Xu [2] in 2006. For biharmonic equation, the a priori/a posteriori error estimate was studied in [3‐6] and the convergence and optimality of the adaptive Morley element method was proved in [7, 8]. The Morley element has been employed to solve the biharmonic eigenvalue problem, including the vibration of a plate; and [9] studied its a priori error estimate. [10, 11] studied a posteriori error estimate and the adaptive method, [12] adopted a new method dispensing with any additional regularity assumption to study the error estimates and adaptive algorithms. This paper further studies the adaptive Morley element method and has the following features:
1.
The adaptive finite element methods, which were first proposed by Babuska and Rheinboldt [13], have gained an extensive attention in academia. More and more researchers entered this field and obtained many good results, most of which have been systemically summarized in [5, 14‐16]. And [10, 12] have employed the adaptive Morley element algorithms for the biharmonic eigenvalue problem based on solving directly the original eigenvalue problem \(a(u,v)=\lambda b(u,v)\) in each iteration. In this paper, we establish the adaptive Morley element algorithms based on the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to solve the biharmonic eigenvalue problem. The shifted-inverse iteration method based on the multigrid discretizations has been studied in-depth (see [17] and the references therein), but they did not involve the Morley element. With our method, the solution of an original eigenvalue problem is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a series of linear algebraic equations on finer and finer grids. Therefore, our method is more efficient than the method in [10, 12].
2.
For fourth order equations in \(\mathbb{R}^{3}\), it is difficult to employ a conforming element. For instance, Zenicek constructed a conforming tetrahedral finite element with 9 degree of polynomials and 220 nodal parameters [5], while the Morley tetrahedral element [2] has only 10 nodal parameters. Based on [4], we comply with the adaptive Morley element computation for the biharmonic eigenvalue problem in \(\mathbb{R}^{3}\). Numerical results indicate that the adaptive algorithms are very efficient.
3.
A family of good adaptive meshes should satisfy \(h=O(h_{\min}^{\alpha})\), where h is the mesh size, \(h_{\min}\) is the diameter of the smallest element, and α is the regularity index of the biharmonic equation over the domain with reentrant corner (see [18]). However, we find through the numerical computation that \(\frac{h}{h_{\min}^{\alpha}}\) will become bigger and bigger when the iteration increases for the standard adaptive algorithm. Thus, referring to [19], we combine the standard local refined adaptive algorithm with uniformly refined algorithm to give new algorithms.
2 Preliminary
Consider the following biharmonic eigenvalue problem:
where \(\Omega\in \mathbb{R}^{n}\) is a polyhedral domain with boundary ∂Ω, \(\frac{\partial u}{\partial\gamma}\) is the outward normal derivative on ∂Ω.
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Let \(H^{s}(\Omega)\) denote a usual Sobolev space with norm \(\Vert \cdot \Vert _{s,\Omega}\) (\(\Vert \cdot \Vert _{s}\)), \(H_{0}^{2}(\Omega)=\{v\in H^{2}(\Omega): v|_{\partial\Omega}=\frac{\partial v}{\partial \gamma}|_{\partial\Omega}=0\}\) with norm \(\Vert \cdot \Vert _{2}\) and semi-norm \(|\cdot|_{2}\).
The weak form of (2.1) is to seek \((\lambda,u)\in \mathbb{R}\times H_{0}^{2}(\Omega)\) with \(u\neq 0\) such that
In the case of \(n=2\), (2.2) is the weak form of clamped plate vibration.
It is easy to verify that \(a(u,v)\) is a symmetric, continuous, and \(H_{0}^{2}(\Omega)\)-elliptic bilinear form. Let \(\Vert u \Vert _{a}=\sqrt{a(u,u)}\), then the norms \(\Vert u \Vert _{a}\), \(\Vert u \Vert _{2}\), and \(|u|_{2}\) are equivalent.
We assume that \(\pi_{h}=\{\kappa\}\) is a regular simplex partition of Ω and satisfies \(\overline{\Omega}=\bigcup\overline{\kappa}\) (see [20]). Let \(h_{\kappa}\) be the diameter of κ, and \(h=\max\{h_{\kappa}: \kappa\in \pi_{h}\}\) be the mesh size of \(\pi_{h}\) (\(h<1\)), \(h_{\min}=\min\{h_{\kappa}: \kappa\in \pi_{h}\}\). Let \(\varepsilon_{h}=\{F\}\) denote the set of faces ((\(n-1\))-simplexes) of \(\pi_{h}\), and let \(\varepsilon_{h}'=\{l\}\) denote the set of faces (\(n-2\))-simplexes of \(\pi_{h}\). When \(n=2\), \(l=z\) is a vertex of κ, and \(\frac{1}{\operatorname{meas}(l)}\int_{l}v=v(z)\). Let \(\pi_{h}(\kappa)\) denote the set of all elements sharing a common face with the element κ. Let \(\kappa_{+}\) and \(\kappa_{-}\) be any two n-simplexes with a face F in common such that the unit outward normal to \(\kappa_{-}\) at F corresponds to \(\gamma_{F}\). We denote the jump of v across the face F by
$$[v]=(v|_{\kappa_{+}}-v|_{\kappa_{-}})|_{F}. $$
And the jump on boundary faces is simply given by the trace of the function on each face.
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In the papers [2, 5], the Morley element space is defined by
The Morley element space \(S^{h}\subset L_{2}(\Omega)\), \(S^{h}\not\subset H^{1}(\Omega)\). Let
$$\begin{aligned} \Vert v \Vert _{m,h}^{2}=\sum _{\kappa\in \pi_{h}} \Vert v \Vert _{m,\kappa}^{2},\qquad \vert v \vert _{m,h}^{2}=\sum _{\kappa\in \pi_{h}} \vert v \vert _{m,\kappa}^{2},\quad m=0,1,2. \end{aligned}$$
From Lemma 8 in [2], we know that \(|\cdot|_{2,h}\) is equivalent to \(\Vert \cdot \Vert _{2,h}\), \(\Vert \cdot \Vert _{2,h}\) is a norm in \(S^{h}\), and \(a_{h}(\cdot,\cdot)\) is a uniformly \(S^{h}\)-elliptic bilinear form, and \(\Vert \cdot \Vert _{h}=a_{h}(\cdot,\cdot)^{\frac{1}{2}}\) is a norm in \(S^{h}\). And the following equality holds for any \(w\in H_{0}^{2}(\Omega)\):
Define the solution operators \(T:L_{2}({\Omega})\to H_{0}^{2}(\Omega)\subset L_{2}({\Omega})\) and \(T_{h}: L_{2}({\Omega})\to S^{h}\) as follows:
$$ \begin{aligned} &a(Tf,v)=b(f,v),\quad \forall v \in H_{0}^{2}( \Omega), \\ &a_{h}(T_{h}f,v)=b(f,v),\quad \forall v \in S^{h}. \end{aligned} $$
(2.5)
Then \(T, T_{h}: L_{2}(\Omega) \to L_{2}(\Omega)\) are self-adjoint and compact.
It is well known that the eigenvalue problem (2.1) has countably many eigenvalues, which are real and positive diverging to +∞. Suppose that λ and \(\lambda_{h}\) are the kth eigenvalue of (2.2) and (2.3), respectively, the algebraic multiplicity of λ is equal to q, \(\lambda=\lambda_{k}=\lambda_{k+1}=\cdots=\lambda_{k+q-1}\). Let \(M(\lambda)\) be the space spanned by all eigenfunctions corresponding to λ and \(M_{h}(\lambda)\) be the direct sum of eigenspaces corresponding to all eigenvalues of (2.3) that converge to λ. Let \(\hat{M}(\lambda)=\{u:u\in M(\lambda), \Vert u \Vert _{h}=1\}\).
The saturation condition was analyzed in [21‐23], especially, it was analyzed in [22] for very general cases. According to this condition, we can make the following assumption:
Due to the generalized Poincare–Friedrichs inequality, Theorem 3 in [24] and \(a_{h}(u-I_{h}u,v)=0, \forall v\in S^{h}\) (see [5]), we deduce for any \(w\in S^{h}, u\in H_{0}^{2}(\Omega)\)
Suppose \(w\in H^{2+r}(\Omega)\), \(r\in (\frac{1}{2},1 ]\), then we have the following estimate:
$$ \bigl\vert E_{h}(w,v) \bigr\vert \leq C_{4} h^{r} \bigl( \Vert w \Vert _{2+r}+h^{2-r} \Vert f \Vert _{0} \bigr) \Vert v \Vert _{h}, \quad \forall v\in S^{h}+H_{0}^{2}(\Omega). $$
(2.9)
Using the trace inequality [5] proves the above estimate under the case \(r=1\). Using the arguments in [5], we can obtain the above estimate under the case \(r=(\frac{1}{2},1]\) (also see [11]).
We can derive the following Lemma 2.1 from Lemma 2.3 in [25].
Lemma 2.1
Letλand\(\lambda_{h}\)be the\(kth\)eigenvalue of (2.2) and (2.3), respectively. Then for any eigenfunction\(u_{h}\)corresponding to\(\lambda_{h}\)with\(\Vert u_{h} \Vert _{h}=1\), there exist\(u\in M(\lambda)\)and\(h_{0}>0\)such that if\(h\leq h_{0}\),
where constants\(C_{5}\)and\(C_{6}\)are positive and only depend onλ.
The following inequality on Rayleigh quotient plays an important role.
Theorem 2.1
Let\((\lambda,u)\)be an eigenpair of (2.2), \(v\in S^{h}\)with\(\Vert v \Vert _{h}=1\)and\(\Vert v-u \Vert _{h}\leq (4 C_{3}\sqrt{\lambda})^{-1}\), then the Rayleigh quotient\(R(v)=\frac{a_{h}(v,v)}{ \Vert v \Vert ^{2}_{0}}\)satisfies
where\(C_{7}=4\lambda (1+\lambda C_{3}^{2} )(4C_{3}\sqrt{\lambda})^{r-1}+\frac{8C_{4}}{C_{1}^{r}}\lambda ( \Vert u \Vert _{2+r}+h^{2-r}\lambda \Vert u \Vert _{0} )\).
Proof
Since \(u\in M(\lambda), v\in S^{h}, \Vert v \Vert _{h}=1\) and \(\Vert v-u \Vert _{h}\leq (4 C_{3}\sqrt{\lambda})^{-1}\), by Lemma 3.1 in [26] we have
3 The shifted-inverse iteration based on multigrid discretization
Let \(\{S^{h_{i}}\}_{0}^{\infty}\) be a family Morley element spaces, \(h_{0}=H\). Refer to the references [17], we present the following calculation schemes.
Scheme 1
(Rayleigh quotient iteration based on multigrid discretizations)
Given the iteration times l.
Step 1. Solve (2.3) on \(S^{H}\): Find \((\lambda_{H},u_{H})\in \mathbb{R}\times S^{H}\) such that \(\Vert u_{H} \Vert _{H}=1\) and
Step 5. If \(i>i_{0}\), then \(\lambda^{h_{i_{0}}}\Leftarrow \lambda^{h_{i-1}}\), \(i\Leftarrow i+1\), turn to Step 6; else, \(i\Leftarrow i+1\), and return to Step 3.
Step 6. Solve a linear system on \(S^{h_{i}}\): Find \(u'\in S ^{h_{i}}\) such that
Step 8. If \(i=l\), then output \((\lambda^{h_{l}},u^{h_{l}})\), stop; else, \(i\Leftarrow i+1\), and return to Step 6.
Strictly speaking, the above \(a_{h}(\cdot,\cdot)\) and \(\Vert \cdot \Vert _{h}\) should be written as \(a_{h_{i}}(\cdot,\cdot)\) and \(\Vert \cdot \Vert _{h_{i}}\). For the sake of simplicity, we write \(a_{h_{i}}(\cdot,\cdot)\) and \(\Vert \cdot \Vert _{h_{i}}\) as \(a_{h}(\cdot,\cdot)\) and \(\Vert \cdot \Vert _{h}\), in this paper.
4 The theoretical analysis
In this section, we will prove the convergence of \((\lambda^{h_{l}},u^{h_{l}})\) derived from Scheme 1/Scheme 2, and that the constants appearing in the error estimates are not only independent of mesh parameter but also iterative times l.
In the following discussion, let \((\lambda_{k},u_{k})\) and \((\lambda_{k,h},u_{k,h})\) denote the \(kth\) eigenpair of (2.2) and (2.3), respectively, and \(\mu_{k}=\frac{1}{\lambda_{k}},\mu_{k,h}=\frac{1}{\lambda_{k,h}},M(\mu_{k})=M(\lambda_{k}),M_{h}(\mu_{k})=M_{h}(\lambda_{k})\).
Our analysis is based on the following Lemma 4.1 (see Lemma 4.1 in [17]).
Lemma 4.1
Let\((\mu_{0},u_{0})\)be an approximation for\((\mu_{k},u_{k})\), where\(\mu_{0}\)is not an eigenvalue of\(T_{h}\), and\(u_{0}\in S^{h}\)with\(\Vert u_{0} \Vert _{h}=1\). Suppose that
where \(\lambda_{0}\) is an approximate eigenvalue of \(\lambda_{k}\), \(u^{h_{l}}\) is an approximate eigenfunction obtained by Scheme 1 or Scheme 2, and ρ is the separation constant of the eigenvalue \(\mu_{k}=\frac{1}{\lambda_{k}}\).
Condition 4.1 plays a key role in proving Theorem 4.1, by which we can prove Theorems 4.2–4.3. In the proof of Theorems 4.2–4.3, we can deduce that Condition 4.1 holds when the mesh size H is appropriately small. However, it is difficult to verify the condition whether the mesh size H is appropriately small or not. And it seems to be a necessary condition in many papers on the convergence and error estimates of the finite element method for eigenvalue problem. But numerical experiments in Sect. 6 present a satisfying practical performance for our algorithms, which shows that it is unnecessary for the mesh size H to be appropriately small, even though the theory is not complete.
The following Theorems 4.1–4.3 are the generalization of Theorems 4.2–4.4 in [17].
Theorem 4.1
Let\((\lambda_{k}^{h_{l}},u_{k}^{h_{l}})\)be an approximate eigenvalue obtained by Scheme1or Scheme 2. Assume that Lemma2.1and Condition4.1hold with\(\lambda_{0}=\lambda_{k}^{h_{l-1}}\)for Scheme1or\(\lambda_{0}=\lambda_{k}^{h_{i0}}\)for Scheme2. Then there exists\(u_{k}\in M(\lambda_{k})\)such that
We use Lemma 4.1 to complete the proof. Select \(\mu_{0}=\frac{1}{\lambda_{0}}\) and \(u_{0}=\frac{\lambda_{k}^{h_{l-1}}T_{h_{l}}u_{k}^{h_{l-1}}}{ \Vert \lambda_{k}^{h_{l-1}}T_{h_{l}}u_{k}^{h_{l-1}} \Vert _{h}}\). Then, by (2.6) and (2.8), we have
Noting that \(\Vert \bar{u} \Vert _{h}\geq \Vert u_{k}^{h_{l-1}} \Vert _{h}- \Vert \bar{u}-u_{k}^{h_{l-1}} \Vert _{h}\geq1-\delta_{0}\geq\frac{1}{2}\), thus, by Lemma 3.1 in [26], we have
Let the eigenvectors \(\{u_{j,h_{l}}\}_{k}^{k+q-1}\) be an orthogonal basis of \(M_{h_{l}}(\lambda_{k})\) with respect to \(a_{h}(\cdot,\cdot)\). Denote
Noting that the constants \(C_{3},C_{5},C_{6},C_{7}\) and ρ are independent of mesh parameters and iterative times l, and \(\Vert u_{k}^{h_{l-1}}-\bar{u} \Vert _{h}\leq\delta_{0}, \vert \lambda_{0}-\lambda_{k} \vert \leq\delta_{0}\) and \(\delta_{h_{l}}(\lambda_{k})\leq\delta_{0}\), by (4.10) and (4.2), we know that there exists a positive constant \(C_{0}\) that is independent of mesh parameters and l such that (4.6) holds. And we can have \(C_{0}\geq C_{5}\). □
We need the following two conditions (see Conditions 4.2 and 4.3 in [17]).
Condition 4.2
There exists \(t_{i}\in(1,2]\) (\(i=1,2,\ldots \)) such that \(\delta_{h_{i}}(\lambda_{k})=\delta_{h_{i-1}}^{t_{i}}(\lambda_{k})\) and \(\delta_{h_{i}}(\lambda_{k})\rightarrow 0\) (\(i\rightarrow\infty\)).
Condition 4.2 is easily satisfied; for example, for smooth eigenfunction, by using the uniform mesh, choose \(h_{0}=\frac{\sqrt{2}}{8}\), \(h_{1}=\frac{\sqrt{2}}{32}\), \(h_{2}=\frac{\sqrt{2}}{64}\), and \(h_{3}=\frac{\sqrt{2}}{128}\); then we have \(h_{i}=h^{t_{i}}_{i-1}\), i.e., \(\delta_{h_{i}}(\lambda_{k})=\delta^{t_{i}}_{h_{i}-1}(\lambda_{k})\), where \(t_{1}\approx1.80,t_{2}\approx1.22,t_{3}\approx1.18\). For a nonsmooth eigenfunction, the condition could be met when the local refinement is done near the singular point.
Condition 4.3
For any given number \(\beta_{0}\in(0,1)\), there exists \(0<\beta_{0}\leq\beta_{i}<1\) (\(i=1,2,\ldots\)) such that \(\delta_{h_{i}}(\lambda_{k})=\beta_{i}\delta_{h_{i-1}}(\lambda_{k}),\delta_{h_{i}}(\lambda_{k})\rightarrow 0\) (\(i\rightarrow \infty\)).
Theorem 4.2
Let\((\lambda_{k}^{h_{l}},u_{k}^{h_{l}})\)be an approximate eigenpair obtained by Scheme1. Suppose that Condition4.2holds, then there exist\(u_{k}\in M(\lambda_{k})\)and\(H_{0}>0\)such that if\(H< H_{0}\), Lemma2.1and the following estimates hold:
The proof is completed by using induction and Theorem 4.1 with \(\lambda_{0}=\lambda_{k}^{h_{l-1}}\). Note that \(\delta_{H}(\lambda_{k})\rightarrow0\), then there is a proper small \(H_{0}>0\) such that if \(H\leq H_{0}\), Lemma 2.1 and the following inequalities hold:
When \(l=1\), we have \((\lambda_{k}^{h_{l-1}},u_{k}^{h_{l-1}})=(\lambda_{k,H},u_{k,H})\); from Lemma 2.1 and (2.12), we know that there exists \(\bar{u}\in M(\lambda_{k})\) such that
and \(\delta_{h_{1}}(\lambda{_{j}})\leq\delta_{0}\) (\(j=k-1,k,k+q,j\neq 0\)), i.e. Condition 4.1 holds. Thus, by Theorem 4.1 and \(2-t_{1}\geq0\) and \(C_{5}\leq C_{0}\) we get
then, owing to (4.13)–(4.14), we have \(\Vert u_{k}^{h_{l-1}}-\bar{u} \Vert _{h} \leq\delta_{0}\) and \(\vert \lambda_{k}^{h_{l-1}}-\lambda_{k} \vert \leq\delta_{0}\) (\(j=k-1,k,k+q, j\neq0\)), i.e. the conditions of Theorem 4.1 hold. Therefore, for l, by (4.6) and (4.14) we deduce
Let\((\lambda_{k}^{h_{l}},u_{k}^{h_{l}})\)be an approximate eigenpair obtained by Scheme2. Suppose that Condition4.2holds for\(i\leq i_{0}\)and Condition4.3holds for\(i>i_{0}\). Then there exist\(u_{k}\in M(\lambda_{k})\)and\(H_{0}>0\)such that if\(H\leq H_{0}\)it holds that
The proof is completed by using induction and Theorem 4.1 with \(\lambda_{0}=\lambda_{k}^{h_{i_{0}}}\). Note that \(\delta_{H}(\lambda_{k})\rightarrow 0\) (\(H\rightarrow0\)), then there is a proper small \(H_{0}>0\) such that if \(H\leq H_{0}\), Lemma 2.1 and the following inequalities hold:
which together with (4.18), we get (4.15). Substituting (4.15) into the inequality (2.12), we get (4.16). □
Remark
For some adaptive local refined grids used usually, (2.9) can be expressed as \(\vert E_{h}(u,v) \vert \leq C_{4}h \Vert v \Vert _{h}\), \(\forall v\in S^{h}+H_{0}^{2}(\Omega)\), therefore r in the theorems of this paper can take 1.
5 Adaptive algorithms
In this section, referring to [10, 17, 27], we present six algorithms. We denote Algorithm 1 in [10] as Algorithm 1 in this paper, and Algorithms 2‐3 are established based on Schemes 1‐2, respectively. Then we combine Algorithms 1‐3 with a uniformly refined algorithm to get Algorithms 1M‐3M, respectively. And the a posterior error estimator in the following algorithms comes from [4], that is
where \(w_{h}\) is the finite element approximate solution of (2.4), \(\tau_{F}\) is the tangential vector and \(\gamma_{F}\) the unit outward normal on \(F\in\varepsilon_{h}\).
In the following algorithms, we have to provide an initial shape regular triangulation \(\pi_{h_{0}}\) and a parameter \(\theta \in (0,1)\). Also, from [10, 11] we know that replacing \(w_{h}\) with \(u_{h}\) and replacing f with \(\lambda_{h}u_{h}\) in (5.1), we can obtain the error estimator of Algorithms 1 and 1M. By Lemma 4.1 we can deduce that replacing \(w_{h}\) with \(u^{h}\) and replacing f with \(\lambda^{h}u^{h}\) in (5.1), we can obtain the error estimator of Algorithms 2–3 and Algorithms 2M–3M.
Algorithm 1
Choose the parameter \(0<\theta<1\).
Step 1. Pick any initial mesh \(\pi_{h_{0}}\).
Step 2. Solve (2.3) on \(\pi_{h_{0}}\) for discrete solution \((\lambda_{h_{0}},u_{h_{0}})\).
Step 3.\(l\Leftarrow0\).
Step 4. Compute the local indicators \(\eta_{h_{l}}(\lambda_{h_{l}}u_{h_{l}},u_{h_{l}},\kappa)\).
Step 5. Construct \(\hat{\pi}_{h_{l}}\in \pi_{h_{l}}\) by Marking strategyE and θ.
Step 6. Refine \(\pi_{h_{l}}\) to get a new mesh \(\pi_{h_{l+1}}\) by procedure Refine.
Step 7. Solve (2.3) on \(\pi_{h_{l+1}}\) for discrete solution \((\lambda_{h_{l+1}},u_{h_{l+1}})\).
Step 8.\(l\Leftarrow l+1\) and go to Step 4.
Algorithm 2
Choose the parameter \(0<\theta<1\).
Step 1. Pick any initial mesh \(\pi_{h_{0}}\).
Step 2. Solve (2.3) on \(\pi_{h_{0}}\) for discrete solution \((\lambda_{h_{0}},u_{h_{0}})\).
Step 8.\(\lambda_{0}\Leftarrow\lambda^{h_{l+1}},l\Leftarrow l+1\) and go to Step 4.
Algorithm 3
Choose the parameter \(0<\theta<1\) and an integer \(i_{0}\).
Step 1∼Step 7. The same as Steps 1–7 of Algorithm 2.
Step 8. If \(l< i_{0},\lambda_{0}\Leftarrow\lambda^{h_{l+1}},l\Leftarrow l+1\) and go to Step 4; else \(l\Leftarrow l+1\), and go to Step 4.
A family of good adaptive meshes should satisfy \(h=O(h_{\min}^{\alpha})\). Hence, we give a bound \(C_{r}\) of \(\frac{h}{h_{\min}^{\alpha}}\). When the rate \(\frac{h}{h_{\min}^{\alpha}}\geq C_{r}\) in the process of Algorithms 1M–3M is running, we refine the mesh uniformly for one time. And thus the following three algorithms are derived.
Algorithm 1M
Choose the parameter \(0<\theta<1\), α, and a bound \(C_{r}\) of \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\).
Step 1∼Step 7. The same as Steps 1–7 of Algorithm 1.
Step 8.\(l\Leftarrow l+1\).
Step 9. If \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\geq C_{r}\), then uniformly refine the mesh \(\pi_{h_{l}}\) to get a new mesh \(\pi_{h_{l+1}}\) and go to Step 7, else go to Step 4.
Algorithm 2M
Choose the parameter \(0<\theta<1\), α, and a bound \(C_{r}\) of \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\).
Step 1∼Step 7. The same as Steps 1–7 of Algorithm 2.
Step 9. If \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\geq C_{r}\), then uniformly refine the mesh \(\pi_{h_{l}}\) to get a new mesh \(\pi_{h_{l+1}}\) and go to Step 7, else go to Step 4.
Algorithm 3M
Choose the parameter \(0<\theta<1\), an integer \(i_{0}\), α, and a bound \(C_{r}\) of \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\).
Step 1∼Step 7. The same as Steps 1–7 of Algorithm 2.
Step 8. If \(l< i_{0},\lambda_{0}\Leftarrow\lambda^{h_{l+1}},l\Leftarrow l+1\); else \(l\Leftarrow l+1\).
Step 9. If \(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\geq C_{r}\), then uniformly refine the mesh \(\pi_{h_{l}}\) to get a new mesh \(\pi_{h_{l+1}}\) and go to Step 7, else go to Step 4.
Marking strategy E
Given parameter \(0<\theta<1\):
Step 1. Construct a minimal subset \(\widehat{\pi}_{h_{l}}\) of \(\pi_{h_{l}}\) by selecting some elements in \(\pi_{h_{l}}\) such that
Step 2. Mark all the elements in \(\widehat{\pi}_{h_{l}}\).
Marking strategy E1
To get Marking strategy E1 we only replace \(\lambda_{h_{l}}\) and \(u_{h_{l}}\) in Marking strategy E with \(\lambda^{h_{l}}\) and \(u^{h_{l}}\), respectively.
Algorithms 1M–3M including steps with uniform refinement seem to be opposite to the adaptive concept. Indeed, the combination of adaptive algorithms and uniform refinement meets the certain mesh-grading properties, thus improving the efficiency of Algorithms 1–3 (see Tables 1–3 in Sect. 6).
Table 1
The smallest eigenvalue solved by Algorithm 1 and Algorithm 1M
l
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}\)
CPU(s)
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}^{M}\)
CPU(s)
1
2945
0.044
0.210
6333.637
0.275
2945
0.044
0.210
6333.637
0.086
2
2957
0.044
0.250
6368.756
0.368
2957
0.044
0.250
6368.756
0.162
3
3035
0.044
0.297
6426.312
0.452
3035
0.044
0.297
6426.312
0.239
4
3135
0.044
0.354
6459.396
0.537
3135
0.044
0.354
6459.396
0.320
5
3345
0.044
0.420
6506.181
0.629
3345
0.044
0.420
6506.181
0.405
6
3609
0.044
0.500
6540.464
0.726
3609
0.044
0.500
6540.464
0.499
7
3979
0.044
0.595
6574.027
0.834
3979
0.044
0.595
6574.027
0.605
8
4459
0.044
0.707
6588.671
0.957
4459
0.044
0.707
6588.671
0.723
9
5097
0.044
0.841
6606.244
1.10
5097
0.044
0.841
6606.244
0.860
10
5787
0.044
1.00
6615.776
1.27
5787
0.044
1.00
6615.776
1.02
11
6665
0.044
1.19
6631.992
1.48
6665
0.044
1.19
6631.992
1.21
12
7791
0.044
1.41
6642.939
1.71
31,697
0.022
0.841
6688.240
2.12
13
9110
0.044
1.68
6656.854
1.97
34,833
0.022
1.00
6690.595
3.15
14
10,591
0.044
2.00
6662.468
2.27
39,191
0.022
1.19
6693.066
4.43
15
12,295
0.044
2.38
6665.980
2.65
173,919
0.011
0.841
6701.061
11.4
16
14,331
0.044
2.83
6671.824
3.10
189,989
0.011
1.00
6701.477
19.4
17
16,641
0.044
3.36
6676.967
3.62
211,977
0.011
1.19
6701.750
29.0
18
19,497
0.044
4.00
6680.844
4.21
948,969
0.006
0.841
6703.147
82.8
19
22,925
0.044
4.76
6684.502
4.92
1,025,149
0.006
1.00
6703.198
141
20
27,171
0.044
5.66
6686.546
5.77
1,131,177
0.006
1.19
6703.244
210
21
32,088
0.044
6.73
6689.797
6.76
5,114,697
0.003
0.841
6703.512
587
22
37,703
0.044
8.00
6692.349
7.95
–
–
–
–
–
23
44,289
0.044
9.51
6694.425
9.39
–
–
–
–
–
24
52,103
0.044
11.3
6695.960
11.1
–
–
–
–
–
25
60,857
0.044
13.5
6696.560
13.2
–
–
–
–
–
26
70,881
0.044
16.0
6697.400
15.7
–
–
–
–
–
27
83,091
0.031
13.5
6698.304
18.7
–
–
–
–
–
28
98,019
0.031
16.0
6699.274
22.7
–
–
–
–
–
29
116,273
0.031
19.0
6699.950
27.4
–
–
–
–
–
30
136,557
0.031
22.6
6700.589
33.1
–
–
–
–
–
31
160,465
0.022
16.0
6701.087
40.0
–
–
–
–
–
32
188,195
0.022
19.0
6701.469
48.2
–
–
–
–
–
33
221,401
0.022
22.6
6701.858
58.0
–
–
–
–
–
34
257,797
0.022
26.9
6702.060
69.7
–
–
–
–
–
35
301,063
0.022
32.0
6702.246
84.6
–
–
–
–
–
36
353,201
0.022
38.1
6702.411
102
–
–
–
–
–
37
416,609
0.022
45.3
6702.557
124
–
–
–
–
–
38
492,039
0.022
45.3
6702.767
151
–
–
–
–
–
39
577,233
0.022
53.8
6702.937
182
–
–
–
–
–
40
677,271
0.016
45.3
6703.035
220
–
–
–
–
–
41
793,765
0.016
53.8
6703.115
266
–
–
–
–
–
42
934,557
0.016
64.0
6703.190
321
–
–
–
–
–
43
1,084,193
0.016
64.0
6703.237
388
–
–
–
–
–
44
1,267,059
0.016
76.1
6703.272
465
–
–
–
–
–
45
1,487,051
0.016
90.5
6703.320
558
–
–
–
–
–
46
1,756,709
0.016
108
6703.362
672
–
–
–
–
–
47
2,065,245
0.011
90.5
6703.407
809
–
–
–
–
–
48
2,420,223
0.011
108
6703.446
973
–
–
–
–
–
49
2,834,373
0.011
128
6703.468
1171
–
–
–
–
–
50
3,319,763
0.011
128
6703.487
1415
–
–
–
–
–
51
3,894,763
0.011
152
6703.505
1706
–
–
–
–
–
52
4,522,239
0.011
181
6703.516
2060
–
–
–
–
–
Table 2
The smallest eigenvalue solved by Algorithm 2 and Algorithm 2M
l
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}^{R}\)
CPU(s)
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}^{\mathrm{RM}}\)
CPU(s)
1
2945
0.044
0.210
6333.637
0.133
2945
0.044
0.210
6333.637
0.090
2
2957
0.044
0.297
6373.503
0.228
2957
0.044
0.297
6373.503
0.140
3
3031
0.044
0.354
6538.971
0.280
3031
0.044
0.354
6538.971
0.192
4
3067
0.044
0.420
6443.737
0.371
3067
0.044
0.420
6443.737
0.250
5
3237
0.044
0.500
6489.761
0.426
3237
0.044
0.500
6489.761
0.305
6
3445
0.044
0.595
6523.466
0.487
3445
0.044
0.595
6523.466
0.365
7
3811
0.044
0.707
6566.620
0.560
3811
0.044
0.707
6566.620
0.431
8
4195
0.044
0.841
6595.431
0.636
4195
0.044
0.841
6595.431
0.504
9
4678
0.044
1.00
6597.955
0.720
4678
0.044
1.00
6597.955
0.588
10
5293
0.044
1.19
6607.441
0.814
5293
0.044
1.19
6607.441
0.709
11
6118
0.044
1.41
6623.248
0.924
25,297
0.022
0.841
6683.573
1.15
12
6997
0.044
1.68
6634.588
1.05
27,723
0.022
1.00
6686.324
1.63
13
8232
0.044
2.00
6648.804
1.20
30,933
0.022
1.19
6688.518
2.38
14
9527
0.044
2.38
6658.106
1.39
139,409
0.011
0.841
6700.069
5.87
15
11,102
0.044
2.83
6663.393
1.59
153,179
0.011
1.00
6700.762
9.71
16
12,928
0.044
3.36
6667.166
1.82
168,897
0.011
1.19
6701.161
15.4
17
15,139
0.044
4.00
6673.763
2.11
740,417
0.006
0.841
6702.983
39.1
18
17,619
0.044
4.76
6678.433
2.43
807,451
0.006
1.00
6703.097
65.4
19
20,763
0.044
5.66
6682.562
2.82
882,597
0.006
1.19
6703.149
103
20
24,365
0.044
6.73
6685.164
3.27
3,907,351
0.003
0.841
6703.486
236
21
28,967
0.044
8.00
6687.944
3.81
4,218,771
0.003
1.00
6703.504
382
22
34,068
0.044
9.51
6690.675
4.50
–
–
–
–
–
23
40,007
0.044
11.3
6692.914
5.39
–
–
–
–
–
24
47,117
0.044
13.5
6694.937
6.46
–
–
–
–
–
25
55,275
0.044
16.0
6696.294
7.64
–
–
–
–
–
26
64,407
0.044
19.0
6696.867
9.09
–
–
–
–
–
27
75,259
0.031
13.5
6697.823
10.8
–
–
–
–
–
28
88,353
0.031
16.0
6698.752
13.2
–
–
–
–
–
29
104,269
0.031
19.0
6699.457
16.1
–
–
–
–
–
30
123,285
0.031
22.6
6700.133
19.4
–
–
–
–
–
31
145,061
0.031
26.9
6700.774
23.3
–
–
–
–
–
32
170,397
0.022
22.6
6701.291
27.9
–
–
–
–
–
33
199,833
0.022
26.9
6701.638
33.5
–
–
–
–
–
34
235,261
0.022
26.9
6701.906
40.0
–
–
–
–
–
35
272,877
0.022
32.0
6702.117
48.6
–
–
–
–
–
36
319,549
0.022
38.1
6702.292
58.8
–
–
–
–
–
37
375,279
0.022
45.3
6702.462
71.1
–
–
–
–
–
38
444,149
0.022
53.8
6702.631
86.1
–
–
–
–
–
39
522,525
0.022
64.0
6702.819
104
–
–
–
–
–
40
613,375
0.022
76.1
6702.964
125
–
–
–
–
–
41
719,217
0.016
53.8
6703.062
149
–
–
–
–
–
42
844,333
0.016
64.0
6703.144
179
–
–
–
–
–
43
988,863
0.016
76.1
6703.208
214
–
–
–
–
–
44
1,150,057
0.016
90.5
6703.256
255
–
–
–
–
–
45
1,346,861
0.016
108
6703.292
304
–
–
–
–
–
46
1,584,041
0.016
128
6703.337
362
–
–
–
–
–
47
1,873,597
0.016
152
6703.378
433
–
–
–
–
–
48
2,196,067
0.011
108
6703.422
509
–
–
–
–
–
49
2,572,281
0.011
128
6703.454
598
–
–
–
–
–
50
3,010,543
0.011
152
6703.474
705
–
–
–
–
–
51
3,538,161
0.011
181
6703.497
831
–
–
–
–
–
52
4,120,833
0.011
215
6703.510
979
–
–
–
–
–
Table 3
The smallest eigenvalue solved by Algorithm 3 and Algorithm 3M
l
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}^{F}\)
CPU(s)
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\frac{h_{l}}{h_{l_{\min}}^{\alpha}}\)
\(\lambda_{1,h_{l}}^{FM}\)
CPU(s)
1
2945
0.044
0.210
6333.637
0.086
2945
0.044
0.210
6333.637
0.086
2
2957
0.044
0.297
6373.503
0.137
2957
0.044
0.297
6373.503
0.137
3
3031
0.044
0.354
6538.971
0.187
3031
0.044
0.354
6538.971
0.187
4
3067
0.044
0.420
6443.737
0.244
3067
0.044
0.420
6443.737
0.245
5
3237
0.044
0.500
6489.761
0.299
3237
0.044
0.500
6489.761
0.300
6
3445
0.044
0.595
6523.466
0.358
3445
0.044
0.595
6523.466
0.360
7
3811
0.044
0.707
6566.620
0.423
3811
0.044
0.707
6566.620
0.426
8
4195
0.044
0.841
6595.431
0.496
4195
0.044
0.841
6595.431
0.498
9
4678
0.044
1.00
6597.955
0.577
4678
0.044
1.00
6597.955
0.581
10
5293
0.044
1.19
6607.441
0.670
5293
0.044
1.19
6607.441
0.696
11
6118
0.044
1.41
6623.248
0.779
25,297
0.022
0.841
6683.573
1.14
12
6997
0.044
1.68
6634.588
0.903
27,723
0.022
1.00
6686.324
1.62
13
8232
0.044
2.00
6648.804
1.05
30,933
0.022
1.19
6688.518
2.37
14
9527
0.044
2.38
6658.106
1.21
139,409
0.011
0.841
6700.069
5.85
15
11,102
0.044
2.83
6663.393
1.41
153,179
0.011
1.00
6701.996
9.71
16
12,928
0.044
3.36
6667.166
1.64
164,253
0.011
1.19
6701.274
15.2
17
15,139
0.044
4.00
6673.763
1.93
715,319
0.006
0.841
6702.994
38.2
18
17,619
0.044
4.76
6678.433
2.25
780,735
0.006
1.00
6703.096
63.8
19
20,763
0.044
5.66
6682.562
2.63
853,934
0.006
1.19
6703.205
100
20
24,365
0.044
6.73
6685.164
3.08
3,774,935
0.003
0.841
6703.503
231
21
28,967
0.044
8.00
6687.944
3.60
4,083,915
0.003
1.00
6703.538
372
22
34,068
0.044
9.51
6690.675
4.21
–
–
–
–
–
23
40,007
0.044
11.3
6692.914
4.97
–
–
–
–
–
24
47,117
0.044
13.5
6694.937
5.89
–
–
–
–
–
25
55,275
0.044
16.0
6696.294
6.98
–
–
–
–
–
26
64,407
0.044
19.0
6696.867
8.28
–
–
–
–
–
27
75,259
0.031
13.5
6697.823
9.85
–
–
–
–
–
28
88,357
0.031
16.0
6698.753
12.0
–
–
–
–
–
29
104,277
0.031
19.0
6699.461
14.6
–
–
–
–
–
30
123,275
0.031
22.6
6700.132
17.7
–
–
–
–
–
31
145,073
0.031
26.9
6700.784
21.4
–
–
–
–
–
32
170,409
0.022
22.6
6701.303
25.9
–
–
–
–
–
33
199,844
0.022
26.9
6701.644
31.2
–
–
–
–
–
34
235,273
0.022
26.9
6701.901
37.6
–
–
–
–
–
35
272,825
0.022
32.0
6702.117
45.7
–
–
–
–
–
36
319,389
0.022
38.1
6702.299
55.5
–
–
–
–
–
37
375,188
0.022
45.3
6702.459
67.4
–
–
–
–
–
38
443,902
0.022
53.8
6702.642
81.7
–
–
–
–
–
39
522,189
0.022
64.0
6702.815
98.8
–
–
–
–
–
40
612,931
0.022
76.1
6702.966
119
–
–
–
–
–
41
718,761
0.016
53.8
6703.061
143
–
–
–
–
–
42
844,127
0.016
64.0
6703.150
171
–
–
–
–
–
43
988,405
0.016
76.1
6703.208
205
–
–
–
–
–
44
1,149,526
0.016
90.5
6703.256
244
–
–
–
–
–
45
1,346,037
0.016
108
6703.295
291
–
–
–
–
–
46
1,583,069
0.016
128
6703.340
347
–
–
–
–
–
47
1,872,353
0.016
152
6703.381
412
–
–
–
–
–
48
2,194,659
0.011
108
6703.425
490
–
–
–
–
–
49
2,570,539
0.011
128
6703.458
580
–
–
–
–
–
50
3,008,669
0.011
152
6703.478
687
–
–
–
–
–
51
3,535,715
0.011
181
6703.498
813
–
–
–
–
–
52
4,118,331
0.011
215
6703.512
962
–
–
–
–
–
6 Numerical experiment
In this section, we compute the smallest eigenvalue of (2.1) on the L-shaped domain \((0,1)^{2}\setminus[\frac{1}{2},1]^{2}\) by Algorithms 1–3 and Algorithms 1M–3M and \((0,1)^{3}\setminus ([0.5,1]\times[0,1]\times[0.5,1] )\) by Algorithms 1–2 to demonstrate the advantages of the adaptive Morley element method based on the inverse-shift iteration for a biharmonic eigenvalue problem. Our programs are compiled on MATLAB2012a under the package of Chen [28] using HP-Z230 workstation with ROM 32G and CPU 3.60 GHz.
We use the command “∖” to solve (5.2) and use the sparse solver \(\operatorname{eigs}(A,B,1,{'}sm{'})\) to solve (2.3) for the smallest eigenvalues. Before showing the results, some symbols need to be explained:
\(\lambda_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 1.
\(\lambda^{R}_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 2.
\(\lambda^{F}_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 3.
\(\lambda^{M}_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 1M.
\(\lambda^{RM}_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 2M.
\(\lambda^{FM}_{h_{l}}\)
the smallest eigenvalue obtained by the lth iteration using Algorithm 3M.
\(N_{\mathrm{dof}}\)
the number of the degree of freedom.
\(\operatorname{CPU}(s)\)
the time CPU runs from the first iteration to the current iteration.
In \(\mathbb{R}^{2}\), the initial mesh \(\pi_{h_{0}}\) is isosceles right triangle subdivision with mesh size \(\frac{\sqrt{2}}{32}\), and we take \(\theta=0.25\), \(C_{r}=1.1\), \(\alpha=\frac{1}{2}\). We fix shift from the 25th and 13th in Algorithm 3 and Algorithm 3M, respectively. The results are shown in Tables 1–3. We depict the error curves of Algorithms 1–3 and Algorithms 1M–3M in Figs. 1–3.
Figure 1
The convergence rates of the smallest eigenvalue from Algorithm 1(a) and Algorithm 1M(b)
Figure 2
The convergence rates of the smallest eigenvalue from Algorithm 2(a) and Algorithm 2M(b)
Figure 3
The convergence rates of the smallest eigenvalue from Algorithm 3(a) and Algorithm 3M(b)
×
×
×
From Tables 1–3, we can get the conclusion that in the case the accurate are almost same, Algorithms 2–3 take about half time of Algorithm 1. In the case the accurate are almost same, Algorithm iM takes about \(\frac{2}{5}\) time of Algorithm i, \(i=1,2,3\).
The smallest eigenvalue of (2.1) is unknown. Therefore, we replace it with an approximate eigenvalue \(\lambda_{1}\approx6703.585\) in \(\mathbb{R}^{2}\) with high accuracy. It is present that the relative error curves of the smallest eigenvalues derived from Algorithms 1–3 and Algorithms 1M–3M on the adaptive meshes in Figs. 1–3, whose slopes are more or less −1, which shows that all the six Morley element adaptive algorithms can get the optimal convergence rate \(O(h^{2})\) in \(\mathbb{R}^{2}\).
In \(\mathbb{R}^{3}\), the initial mesh \(\pi_{h_{0}}\) is tetrahedron subdivision with mesh size \(\frac{\sqrt{3}}{16}\), and we take \(\theta=0.25\) and \(\lambda_{1}\approx8290.011\) with high accuracy replacing the accurate eigenvalue. It is present that the refined mesh and the relative error curves of the smallest eigenvalues derived from Algorithms 1–2 in Fig. 4, from which we see that Algorithm 2 is more efficient than Algorithm 1, but meanwhile we also see from Table 4 that the mesh size has no change. Because \(N_{\mathrm{dof}}\) in \(\mathbb{R}^{3}\) increases very fast after uniform refinement, which leads to surpassing computer’s memory, we cannot employ Algorithms 1M–3M to solve (2.1).
Figure 4
The refined mesh for the L-shaped domain (a) and the convergence rates of the smallest eigenvalue from Algorithm 1 and Algorithm 2(b) in \(\mathbb{R}^{3}\)
Table 4
The smallest eigenvalue solved by Algorithm 1 and Algorithm 2
l
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\lambda_{1,h_{l}}\)
CPU(s)
\(N_{\mathrm{dof}}\)
\(h_{l}\)
\(\lambda_{1,h_{l}}^{R}\)
CPU(s)
1
54,896
0.108
7547.686
8.25
54,896
0.108
7547.686
9.71
2
57,176
0.108
7612.980
16.5
57,176
0.108
7613.142
13.0
3
60,822
0.108
7678.863
25.6
60,822
0.108
7678.902
16.7
4
67,192
0.108
7736.644
35.1
67,192
0.108
7736.699
20.9
5
76,278
0.108
7802.128
46.5
76,316
0.108
7802.062
25.8
6
86,838
0.108
7871.038
58.6
86,905
0.108
7872.137
31.2
7
101,261
0.108
7949.987
73.8
101,368
0.108
7950.349
38.4
8
121,408
0.108
8001.974
92.4
121,563
0.108
8002.931
47.3
9
146,456
0.108
8041.421
118
146,215
0.108
8040.738
58.1
10
180,528
0.108
8082.211
155
180,203
0.108
8082.247
72.3
11
224,755
0.108
8108.734
211
224,288
0.108
8108.530
92.1
12
282,583
0.108
8141.177
295
281,953
0.108
8141.009
119
13
355,133
0.108
8174.424
414
354,598
0.108
8174.437
156
14
451,162
0.108
8199.573
583
450,739
0.108
8199.502
205
15
561,904
0.108
8220.566
847
561,090
0.108
8220.433
269
16
693,222
0.108
8240.310
2368
691,963
0.108
8240.129
354
17
863,420
0.108
8258.156
9957
861,795
0.108
8258.042
469
18
–
–
–
–
1,084,848
0.108
8272.888
624
19
–
–
–
–
1,357,830
0.108
8281.060
851
20
–
–
–
–
1,730,050
0.108
8290.011
1206
×
Acknowledgements
This work is supported by the Science and Technology Foundation of Guizhou Province of China (Grant nos. LH [2014] 7061 and LKS [2013] 06). We appreciate editors and reviewers for constructive suggestions and helpful comments.
Competing interests
The authors declare to have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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