A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue
- Jing An^{1},
- Hai Bi^{1} and
- Zhendong Luo^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-016-1158-1
© An et al. 2016
Received: 12 July 2016
Accepted: 31 August 2016
Published: 7 September 2016
Abstract
In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order Steklov equation with boundary eigenvalue. By making use of the spectral theory of compact operators and the error formulas of projective operators, we first obtain the error estimates of approximative eigenvalues and eigenfunctions. Then we build a suitable set of basis functions included in \(H^{1}_{0}(\Omega)\cap H^{2}(\Omega)\) and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results.
Keywords
MSC
1 Introduction
Increasing attention has recently been paid to numerical approximations for Steklov equations with boundary eigenvalue, arising in fluid mechanics, electromagnetism, etc. (see, e.g., [1–11]). However, most existing work usually treated the second-order Steklov equations with boundary eigenvalue and there are relatively few articles treating fourth-order ones. The fourth-order Steklov equations with boundary eigenvalue also have been used in both mathematics and physics, for example, the main eigenvalues play a very key role in the positivity-preserving properties for the biharmonic-operator \(\triangle ^{2}\) under the border conditions \(w=\triangle w-\chi w_{\nu}=0\) on ∂Θ (see [12]).
A conforming finite element method was first studied and a bound for the exact eigenvalues was provided in [13] for (1.1)-(1.3). Compared with finite element methods, spectral methods have the characteristics of high accuracy (see [14]). Especially, as far as we know, there is no work at all on spectral methods for the fourth-order Steklov equations with boundary eigenvalue, which is different from the equation and technique in [15]. So, this article aims to build a successful spectral-Galerkin formulation for the fourth-order Steklov equation with boundary eigenvalue. The article includes at least the following three features.
(1) We adopt the generalized Jacobian polynomial to deduce in detail the error formula of the high dimensional projective operator associated with the fourth-order Steklov equation with boundary eigenvalue. Then by employing the spectral method of compact operators, we obtain the satisfactory error formulas of approximative eigenvalues and eigenfunctions.
(2) We formulate a suitable set of basis functions and build the matrix formulation for the discrete variational scheme by means of the tensor product. Especially, we combine a set of basis function in spectral space included in \(H_{0}^{2}(\Theta)\) with two basis functions in spectral space included in \(H_{0}^{1}(\Theta)\) to formulate the basis functions. In this way, the matrix formulation obtained is sparse so that it can easily and efficiently be solved.
(3) We not only present the numerical example in the two-dimensional domain for the equation, but we also provide the three-dimensional one which has not been reported as far as we know. The numerical results explain the effectiveness of our approach.
The remainder of this article is arranged as follows. Section 2 provides some preparations. Section 3 provides the error formulas of the projective-operator and the spectral-Galerkin approximate solutions. In Section 4, we establish the matrix model based on the tensor product for the discrete variational model such that it can easily be solved. In Section 5, we enumerate some numerical examples to confirm the accuracy and efficiency of the theoretical results. Finally, in Section 6, we give main conclusions.
2 Some preparations
Let \(H^{s}(\Theta)\) and \(H^{s}(\partial\Theta)\) be the standard Sobolev space on Θ and ∂Θ with integer order s, respectively. \(H^{0}(\Theta)=L^{2}(\Theta), H^{0}(\partial\Theta )=L^{2}(\partial\Theta), H_{0}^{1}(\Theta)=\{v\in H^{1}(\Theta ):v=0 \mbox{ on } \partial\Theta\}\). The norm in \(H^{s}(\Theta)\) and \(H^{s}(\partial\Theta)\) are represented by \(\Vert \cdot \Vert _{s}\) and \(\Vert \cdot \Vert _{s,\partial\Theta}\), respectively. Throughout this article, C is a generic positive constant independent of the degree N of polynomials that may be unequal at the various places.
Put \(V=H^{2}(\Theta)\cap H^{1}_{0}(\Theta)\). The variational formulation for (1.1)-(1.3) is stated as follows.
The source equation corresponding to (2.1) is denoted in the following form.
Lemma 2.1
Proof
3 Error analysis
We will devote this section to estimating the errors between \(\chi _{N}\) and χ, and \(w_{N}\) and w.
Lemma 3.1
Proof
Theorem 3.1
Proof
Let \(M(\chi)\) represent the eigenfunction subspace with respect to the eigenvalue χ of (2.1).
Theorem 3.2
Proof
The following lemma is directly obtained from Theorem 8.1 and Remark 8.14 in [14].
Lemma 3.2
Before we proceed with the error formulas, it is important to make the following observation: \((\nabla(\tilde{\Pi}_{N}^{\boldsymbol {k},\boldsymbol {l}}w-w),\nabla\upsilon_{N})=-(\tilde {\Pi}_{N}^{\boldsymbol {k},\boldsymbol {l}}w-w,\Delta\upsilon_{N}) =-(\tilde{\Pi}_{N}^{\boldsymbol {k},\boldsymbol {l}}w-w, \boldsymbol {w}^{\boldsymbol {k},\boldsymbol {l}}\Delta\upsilon _{N})_{{\boldsymbol {w}^{\boldsymbol {k},\boldsymbol {l}}}}=0\) (\(\forall\upsilon_{N}\in X_{N}\)). Thus, \(\tilde{\Pi}_{N}^{\boldsymbol {k},\boldsymbol {l}}\) is also an orthometric projection from V into \(X_{N}\).
Theorem 3.3
Proof
4 Efficient implementation of the spectral-Galerkin solutions
We will devote this section to providing the approach to solve equation (2.5) efficiently. To this end, we first formulate a basis of normal orthometric functions for \(X_{N}\).
Let \(\phi_{k}(x)=\frac{1}{\sqrt{2(2k+3)^{2}(2k+5)}}(L_{k}(x)-\frac {2(2k+5)}{2k+7}L_{k+2}(x)+\frac{2k+3}{2k+7}L_{k+4})\) (\(k=0,1,\ldots ,N-4\)), \(\phi_{N-3}(x)=L_{0}(x)-L_{2}(x)\), and \(\phi _{N-2}(x)=L_{1}(x)-L_{3}(x)\). It is obvious that \(\{\phi_{k}\}_{k=0}^{N-2}\) constitutes a basis for \(X_{N}\).
From Lemma 3.1 in [18] and the orthogonality of Legendre polynomial, we immediately obtain the following lemma.
Lemma 4.1
If \(a_{kj}=(\phi_{j}'',\phi_{k}'')\), \(b_{kj}=(\phi _{j},\phi_{k})\), and \(c_{kj}=(\phi_{j}',\phi_{k}')\), then we have the following results.
From the fact that \(\varphi_{i}'(\pm1)=0\) \((i=0,1,\ldots,N-4)\), \(\varphi_{N-3}'(\pm1)=\mp3\), and \(\varphi_{N-2}'(\pm1)=-5\), we immediately obtain the following lemma.
Lemma 4.2
Next we will build the matrix formulation for the discrete equation (2.5) by means of the tensor product.
• Case \(d=2\).
• Case \(d=3\).
Based on (4.2) and (4.4) we can efficiently compute the approximative eigenvalues of (1.1)-(1.3) on the rectangle domain and cubic one, respectively.
5 Numerical examples
The approximative eigenvalues on \(\pmb{\bar{\Theta}=[0,\pi/2]^{2}}\)
N | DOF | \(\boldsymbol {\chi_{1}}\) | \(\boldsymbol {\chi_{2}}\) | \(\boldsymbol {\chi_{4}}\) |
---|---|---|---|---|
15 | 196 | 2.212697396 | 4.416172185 | 6.081723134 |
20 | 361 | 2.212697395 | 4.416172142 | 6.081722889 |
25 | 576 | 2.212697395 | 4.41617214 | 6.081722871 |
30 | 841 | 2.212697395 | 4.41617214 | 6.081722871 |
The approximative eigenvalues on \(\pmb{\bar{\Theta}=[0,\pi/2]^{3}}\)
N | DOF | \(\boldsymbol {\chi_{1}}\) | \(\boldsymbol {\chi_{2}}\) | \(\boldsymbol {\chi_{5}}\) |
---|---|---|---|---|
10 | 729 | 2.989297949 | 4.966408489 | 6.533085863 |
15 | 2,744 | 2.989297401 | 4.966394295 | 6.532973331 |
20 | 6,859 | 2.989297388 | 4.966394153 | 6.532972744 |
25 | 13,824 | 2.989297388 | 4.966394148 | 6.532972712 |
We can see that the eigenvalues in Table 1 have at least ten-digit accuracy with \(N= 25\), i.e., \(\mathrm{DOF}=576\). As a comparison, we observe that the eigenvalues obtained by using finite element in [13] only have four-digit accuracy for \(h={\sqrt{2}\pi }/({80})\), i.e., \(\mathrm{DOF}=6{,}244\).
It also can be seen from Table 2 that the eigenvalues have at least eight-digit accuracy with \(N=20\). By computing, for the problem (2.1) on \(\bar {\Theta}=[0,\pi/2]^{3}\), we also see that \(\chi_{1}\) is a simple eigenvalue, \(\chi_{2}\) and \(\chi_{5}\) are all eigenvalues with multiplicity 3.
6 Conclusions
In this study, we have establish an efficient spectral-Galerkin formulation for the fourth-order Steklov equation with boundary eigenvalue. By analyzing the error formulas of projective operators and adopting the compact-operator spectral method, we have derived the error formulas of approximative eigenvalues and eigenfunctions. Then we have formulated a suitable set of basis functions included in \(H^{1}_{0}(\Theta)\cap H^{2}(\Theta)\) and built the matrix formulation for the discrete variational scheme by means of the tensor product. In this way, we can efficiently solve the discrete system and obtain highly accurate approximative eigenvalues. We have provided the numerical examples in rectangle domain and cubic one and the satisfactory results obtained show that our method is very effective. In this study, we confined our focus to the cases in the rectangle domain and the cubic one. In fact, the technique used in this article could be expanded to more general domains by adopting a spectral-element technique.
Declarations
Acknowledgements
This research was supported by National Science Foundation of China grant 11271127, 11671106, and 11661022.
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.
Authors’ Affiliations
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