A high accuracy numerical method based on spectral theory of compact operator for biharmonic eigenvalue equations
- Zhendong Luo^{1}Email author
https://doi.org/10.1186/s13660-016-1014-3
© Luo 2016
Received: 1 February 2016
Accepted: 5 February 2016
Published: 29 February 2016
Abstract
In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base functions, the discrete model with sparse mass and stiff matrices is established so that it is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. Some numerical examples are provided to validate the theoretical results.
Keywords
MSC
1 Introduction
The biharmonic eigenvalue equations are used to describe the vibration and buckling of plates in mechanics (see, e.g., [1–3]) and transmission eigenvalue problem in inverse scattering theory (see, e.g., [4, 5]). Therefore, the biharmonic eigenvalue equations have very wide extensive applications.
However, most of the existing work were concerned with the second-order elliptic eigenvalue problems and there are relatively few articles treating the biharmonic eigenvalue problems. In recent years, the numerical methods for the biharmonic eigenvalue problems adopted mainly the conforming finite element method (see [6–9]), the nonconforming finite element method (see [3, 10–12]), and the mixed finite element method (see [13–16]). For the conforming finite elements method, it requires globally continuously differentiable finite element spaces, therefore, they are difficult to construct and implement (in particular for three-dimensional problems). For the nonconforming finite element method, a disadvantage is that such elements do not come in a natural hierarchy and existing nonconforming elements only involve low-order polynomials so that they are not efficient for capturing smooth solutions. For the mixed finite element method, it can result in spurious solutions on non-convex domains for the boundary conditions of simply supported plates. In the last decade, the \(C^{0}\) interior penalty Galerkin (\(C^{0}\) IPG) method had been developed for second-order elliptic eigenvalue problems (see [17–19]). Recently, Brenner et al. in [20] extended the \(C^{0}\) IPG method to biharmonic eigenvalue problems and provided the method converges for all three types of boundary conditions (see [20]).
However, all methods mentioned above are low-order finite element methods so that it is very difficultly and expensively to obtain high accuracy numerical solutions, especially for the three-dimensional spherical domain. To the best of our knowledge, there is not any article on a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on the spherical domain. Therefore, the task of this paper is to develop a high precision numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations in the spherical domain.
The rest of this paper is organized as follows. Section 2 provides some preliminaries. In Section 3, by employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, we derive the error estimates of approximate eigenvalues and eigenfunctions. In Section 4, by adopting orthogonal spherical base functions, we establish the discrete model with sparse mass and stiff matrices which is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. In Section 5, we provide some numerical examples to validate that the theoretical results are correct. Finally, we provide some conclusions in Section 6.
2 Some preliminaries
The Sobolev spaces and norms used in this paper are standard ([21]). For example, \(H^{s}(\Omega)\) denotes the usual Sobolev space on Ω with real order s, equipped with the norm \(\Vert \cdot \Vert _{s}\), and \(H^{0}(\Omega)=L^{2}(\Omega)\), \(H_{0}^{2}(\Omega)=\{u\in H^{2}(\Omega):u=\frac{\partial u}{\partial \mathbf{n}}=0 \mbox{ on } \partial\Omega\}\).
Moreover, we denote the sets of all nonnegative integers and all real numbers by N and R, respectively. Further, for \(n\in \mathbf {N}\), we denote the collection of all polynomials in d variables with a total degree ≤n by \(\Pi_{n}^{d}\).
Theorem 2.1
3 The operator formulations and error estimates
3.1 Operator formulations
Put \(V=H_{0}^{2}(\Omega)\). Then the weak form of (1)-(2) is given as follows.
The source problem associated with (5) is written as follows.
For the above operator T, we have the following result.
Lemma 3.1
The operator \(T:V\mapsto V\) is a self-adjoint compact one.
Proof
Let E be the bounded set in V. Since V is compactly embedded in \(L_{2}(\Omega)\), so E is the sequentially compact set in \(L^{2}(\Omega )\). From (9), we know that TE is the sequentially compact set in V. Thus, \(T:V\rightarrow V\) is a compact operator. □
Let \(X_{N}=\Pi_{N}^{d} \cap H_{0}^{2}(\Omega)\), then the spectral-Galerkin approximation of (5) is given as follows.
The source problem associated with (10) is written as follows.
We have the following result.
Lemma 3.2
Proof
3.2 Error estimates
In the following, we provide the error estimates. We first provide the following lemma.
Lemma 3.3
Proof
Theorem 3.1
Proof
Let \(M(\lambda)\) denote the eigenfunctions space of (5) corresponding to the eigenvalue λ. We have the following results (also see the proof in [25]).
Theorem 3.2
Proof
Theorem 3.3
Proof
4 Matrix formulation of the spectral-Galerkin approximation
Lemma 4.1
5 Numerical experiments
The approximate eigenvalues on the unit disk \(\pmb{\mathbf {B}^{2}}\) in \(\pmb{\mathbf {R}^{2}}\)
N | \(\boldsymbol{\lambda_{1}}\) | \(\boldsymbol{\lambda_{2}}\) | \(\boldsymbol{\lambda_{4}}\) |
---|---|---|---|
10 | 104.3631056 | 452.0074329 | 1216.451916 |
15 | 104.3631056 | 452.0045101 | 1216.407600 |
20 | 104.3631056 | 452.0045101 | 1216.407600 |
The approximate eigenvalues on unit ball \(\pmb{\mathbf {B}^{3}}\) in \(\pmb{\mathbf {R}^{3}}\)
N | \(\boldsymbol{\lambda_{1}}\) | \(\boldsymbol{\lambda_{2}}\) | \(\boldsymbol{\lambda_{4}}\) |
---|---|---|---|
10 | 237.7210683 | 769.9765197 | 1818.29231 |
15 | 237.7210675 | 769.9634832 | 1818.167926 |
20 | 237.7210675 | 769.9634832 | 1818.167924 |
25 | 237.7210675 | 769.9634832 | 1818.167924 |
It is easily seen from Table 1 that the results have at least ten-digit accuracy with \(N\geq 15\). From numerical results, it is shown that \(\lambda_{1}\) is a simple eigenvalue, but \(\lambda_{2}\) and \(\lambda _{4}\) are all eigenvalues with multiplicity 2.
It is easily seen from Table 2 that the results have at least ten-digit accuracy with \(N\geq20\). From numerical results, we know that \(\lambda_{1}\) also is a simple eigenvalue, but \(\lambda_{2}\) and \(\lambda_{5}\) are eigenvalues with multiplicity 3 and 5, respectively.
By comparing the relative errors of numerical solutions, we found that the convergence rates of numerical solutions approximate \(O(10^{-12})\) when \(N\ge20\), which is consistent with those obtained for the theoretical case. It is also shown that finding the approximate solutions for the biharmonic eigenvalue equations with spectral-Galerkin method is computationally very effective.
6 Conclusions
In this study, we have developed a high accuracy numerical method by means of the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, we have derived the error estimates of approximate eigenvalues and eigenfunctions. We also, respectively, provided two numerical experiments in the two-dimensional case and three-dimensional cases to verify that our method is very effective. While we have restricted our attention in this study to the cases of biharmonic eigenvalue problems with constant coefficient. Whereas the approach presented in this article can be extended to biharmonic eigenvalue problems with variable coefficients.
Declarations
Acknowledgements
This research was supported by National Science Foundation of China grant 11271127.
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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