Research Article | Open Access
Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem
This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001), a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.
Since Steklov eigenvalue problems have important physical background and many applications, for instance, they appear in the analysis of stability of mechanical oscillators immersed in a viscous fluid (see  and the references therein), in the analysis of the antiplane shearing on a system of collinear faults under slip-dependent friction law (see ), in the study of surface waves (see ), in the study of the vibration modes of a structure in contact with an incompressible fluid (see ) and vibration of a pendulum (see ), and in the eigen-oscillations of mechanical systems with boundary conditions containing frequency , and thus, numerical methods for solving Steklov eigenvalue problems have received increasing attention in recent years. Bramble and Osborn  studied the Galerkin approximation of a Steklov eigenvalue problem of nonself-adjoint second order elliptic operators in smooth domain, Andreev and Todorov  discussed the isoparametric finite element method for the approximation of the Steklov eigenvalue problem of second-order self-adjoint elliptic differential operators, Armentano and Padra  introduced and analyzed the conforming linear finite element approximation of the Steklov eigenvalue problem in a bounded polygonal domain, Alonso and Russo , Yang et al. , and Li et al.  studied nonconforming finite elements approximation of the Steklov eigenvalue problem, Li and Yang  and Bi and Yang  discussed a two-grid method of the conforming and non-conforming finite element method, respectively, Li et al.  studied the extrapolation and superconvergence of the Steklov eigenvalue problem, Tang et al.  studied the boundary element approximation, and Cao et al.  discussed multiscale asymptotic method for Steklov eigenvalue equations in composite medias. However, to the best of our knowledge, there have been no reports on spectral method for Steklov eigenvalue problems.
Spectral method is an important numerical method for solving differential equations developed after finite-difference and finite-element methods. Because spectral method has the characteristics of superior accuracy, it is widely used in the field of meteorology, physics, and mechanics (see [18, 19]). This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto (LGL for short) points for solving the Steklov eigenvalue problem. We use spectral approximation theory (see ) to give a priori error estimates. Based on the work of , we discuss a posterior error estimator of the residual type. Moreover, inspired by the work of  this paper combines the shifted-inverse iterative method and spectral method to propose an efficient computation scheme. Finally, by using program with Matlab, we implement the numerical experiments and get satisfactory results.
The rest of the paper is organized as follows. In Section 2, some preliminaries needed in this paper are presented. In Section 3, we discuss a priori and a posterior error estimates for spectral method. In Section 4, we establish an efficient scheme combining the shifted-inverse iteration and spectral method. In Section 5, the numerical experiments on the square domain are reported.
We consider the following boundary value problem: where is a rectangular domain and is the outward normal derivative on . We denote the Sobolev space with norm and seminorm by . Let and be the norms in the space and , respectively.
Throughout this paper, denotes a generic positive constant independent of the polynomial degrees , which may not be the same at each occurrence.
The variational problem associated with (1) is given by the following.
Find , such that where Obviously, the bilinear form is continuous and -elliptic, that is, there exists constants and independent of , , such that
From the Lax-Milgram theorem we know that there exists a unique solution to (2).
The following regularity results (see ) will be needed in the sequel: for , the solution of problem (1) , and furthermore, if , then and where , are constants independent of , and if is convex and if is concave (with being the largest inner angle of ).
The spectral-Galerkin approximation to (2) is the following.
Find , such that
For and each integer , we define the interpolation operator with the LGL interpolation nodes in . We quote from  the following interpolation estimates for spectral method at the tensor-product LGL points
3. Spectral Method and Its Error Estimates for the Steklov Eigenvalue Problem
3.1. Spectral Method for the Steklov Eigenvalue Problem
We consider the following Steklov eigenvalue problem: where is a rectangular domain.
The variational problem associated with (13) is given by the following.
Find , , such that
The spectral-Galerkin approximation to (14) is as the follows.
Find , , such that
3.2. A Prior Error Estimates of Spectral Method for the Steklov Eigenvalue Problem
Let be an eigenpair of (14), let be the space spanned by all eigenfunctions corresponding to the eigenvalue , and let be the direct sum of the eigenspace corresponding to the eigenvalues of (15) that converge to .
Denote where characterizes the degree of the space approximating , and characterize the degrees of approximating the generalized solution of (2).
Lemma 1. Let ; then
Theorem 2. Let ; then let and ; then there exists , such that let and ; then there exists , such that
3.3. A Posterior Error Estimates for the Steklov Eigenvalue Problem
Theorem 2.1 in  gave the following results.
Lemma 3. There exists a positive constant and a linear bounded operator , such that for all and an arbitrary edge of , and there holds
Analysis of the error indicators requires polynomial inverse estimates in weighted Sobolev spaces. Define on the interval the weight function ; then there holds the following polynomial inverse estimates.
Lemma 4. Let , ; then there exists , , and such that for every univariate polynomial of degree , there holds
Proof. See Lemma 2.4 in .
Consider the domain and the weight function . From  we have the two-dimensional polynomial inverse estimates.
Lemma 5. Let , ; then there exists , , and such that for all polynomials there hold
Proof. See Lemma 2.5 in .
Lemma 6. Let , set , and let be defined as the same as , that is, . Then there exists , such that for every and every univariate polynomial of degree , there exists an extension , such that the following holds:
Proof. See Lemma 2.6 in .
Define the global error indicator , , which is expressed as the sum of two terms: where the first term is the weighted internal residual given by and the second term is the weighted edge residual given by where the weight functions and are scaled transformations of the weight functions and , respectively.
By using the proof of , we can get the following results.
Theorem 8. Let . Then there exists independent of , such that
Proof. We denote where is an interpolation operator satisfying Lemma 3. We derive from -ellipticity of that Since thus By using Cauchy-Schwartz inequality, the trace theorem, and Lemma 3, we get Plugging (47) into (46) gives Setting , in (32) and (35), respectively, we obtain and , and thus The proof is completed.
Lemma 9. Let . Then there exists independent of , such that
Proof. Denote , ; then
Now, we consider . Using (35)-(36), we find for that noticing that (35) is valid since the assumption implies , hence we take , in (35) and the above third inequality holds. By the definition , we obtain To get an upper bound of the error indicator in the case of , using (35) we find for that Therefore, We obtain the desired result immediately.
Lemma 10. Let , . Then there exists independent of , such that
Proof. In order to obtain an upper bound for the error indicator , we will use weight function and a proper extension operator. Let be an edge of ; we construct a function and .
Let . We extend to by using Lemma 6 where the polynomial corresponds to , and from (37) we know that is zero on . We extend to such that on ; then Now we estimate , , in turn. First, In the case of we have Combining (58) and (59), we get By the definition of and setting in Lemma 9, we have that Setting yields the assertion for in (61).
For the case , setting and using (35), we find therefore, From the above discussion we complete the proof.
Theorem 11. Let . Then there exists independent of , such that
4. Spectral Method Based on the Shifted-Inverse Iteration
High efficient finite element schemes have been researched further in recent years; for example, see [22, 25–31]. Based on these works, this paper combines the shifted-inverse iteration and spectral method to propose an efficient scheme.
Scheme 13 (spectral method based on the shifted-inverse iteration).
Step 1. Solve (13) in : Find , , such that
Step 2. Solve a linear system in . Find , such that
Step 3. Compute the Rayleigh quotient
5. Numerical Experiments
Consider the Steklov eigenvalue problem (13) where .
We use spectral method with tensor-product nodal basis at the LGL points (LGL spectral method) to solve (13). We construct two-dimensional basis functions by using the tensor product of one-dimensional basis functions. More precisely, let be the Lagrange basis polynomials associated with the LGL points; then , and let ; then (15) is equivalent to the following linear system: where is called stiffness matrix, is called mass matrix, and We program with Matlab to solve (72), and the numerical results are presented in Table 1.
In the tables in this section, DOF denotes the degree of freedom (number of nodes).
From Table 1 we can see that
From Table 2 we see that
Since we do not know the exact eigenvalues of the Steklov eigenvalue problem, many research studies reported the approximate eigenvalues; for example, see [15, 22]. This paper uses the LGL spectral method to solve (13), and in Table 1 we give We think these approximate eigenvalues have high accuracy.
This work is supported by the National Natural Science Foundation of China (Grant no. 11201093).
- C. Conca, J. Planchard, and M. Vanninathan, Fluids and Periodic Structures, vol. 38, John Wiley & Sons, New york, NY, USA, 1995.
- D. Bucur and I. R. Ionescu, “Asymptotic analysis and scaling of friction parameters,” Zeitschrift für Angewandte Mathematik und Physik, vol. 57, no. 6, pp. 1042–1056, 2006.
- S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, NY, USA, 1953.
- A. Bermúdez, R. Rodríguez, and D. Santamarina, “A finite element solution of an added mass formulation for coupled fluid-solid vibrations,” Numerische Mathematik, vol. 87, no. 2, pp. 201–227, 2000.
- H. J. Ahn, “Vibrations of a pendulum consisting of a bob suspended from a wire: the method of integral equations,” Quarterly of Applied Mathematics, vol. 39, no. 1, pp. 109–117, 1981.
- D. B. Hinton and J. K. Shaw, “Differential operators with spectral parameter incompletely in the boundary conditions,” Funkcialaj Ekvacioj, vol. 33, no. 3, pp. 363–385, 1990.
- J. H. Bramble and J. E. Osborn, “Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators,” in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz, Ed., Academic Press, San Diego, Calif, USA, 1972.
- A. B. Andreev and T. D. Todorov, “Isoparametric finite-element approximation of a Steklov eigenvalue problem,” IMA Journal of Numerical Analysis, vol. 24, no. 2, pp. 309–322, 2004.
- M. G. Armentano and C. Padra, “A posteriori error estimates for the Steklov eigenvalue problem,” Applied Numerical Mathematics, vol. 58, no. 5, pp. 593–601, 2008.
- A. Alonso and A. D. Russo, “Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 177–197, 2009.
- Y. Yang, Q. Li, and S. Li, “Nonconforming finite element approximations of the Steklov eigenvalue problem,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2388–2401, 2009.
- Q. Li, Q. Lin, and H. Xie, “Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations,” Applications of Mathematics, vol. 58, no. 2, pp. 129–151, 2013.
- Q. Li and Y. Yang, “A two-grid discretization scheme for the Steklov eigenvalue problem,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 129–139, 2011.
- H. Bi and Y. Yang, “A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9669–9678, 2011.
- M. Li, Q. Lin, and S. Zhang, “Extrapolation and superconvergence of the Steklov eigenvalue problem,” Advances in Computational Mathematics, vol. 33, no. 1, pp. 25–44, 2010.
- W. Tang, Z. Guan, and H. Han, “Boundary element approximation of Steklov eigenvalue problem for Helmholtz equation,” Journal of Computational Mathematics, vol. 16, no. 2, pp. 165–178, 1998.
- L. Cao, L. Zhang, W. Allegretto, and Y. Lin, “Multiscale asymptotic method for Steklov eigenvalue equations in composite media,” SIAM Journal on Numerical Analysis, vol. 51, no. 1, pp. 273–296, 2013.
- C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Berlin, Germany, 2007.
- J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, analysis and applications, vol. 41, Springer, Heidelberg, Germany, 2011.
- I. Babuska and J. Osborn, “Eigenvalue Problems,” in Handbook of Numerical Analysis, Finite Element Methods (Part 1), P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 640–787, Elsevier Science Publishers, 1991.
- J. M. Melenk and B. I. Wohlmuth, “On residual-based a posteriori error estimation in -FEM,” Advances in Computational Mathematics, vol. 15, no. 1–4, pp. 311–331, 2001.
- Y. Yang and H. Bi, “Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 49, no. 4, pp. 1602–1624, 2011.
- Y. Yang, Finite Element Methods for Eigenvalue Problems, Science Press, Beijing, China, 2012.
- J. M. Melenk, “-interpolation of nonsmooth functions and an application to -a posteriori error estimation,” SIAM Journal on Numerical Analysis, vol. 43, no. 1, pp. 127–155, 2005.
- Y. Yang and W. Jiang, “Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem,” Science China Mathematics, vol. 56, no. 6, pp. 1313–1330, 2013.
- J. Hu and Y. Huang, “The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues,” Science China Mathematics, vol. 55, no. 1, pp. 187–196, 2012.
- A. Naga and Z. Zhang, “Function value recovery and its application in eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 50, no. 1, pp. 272–286, 2012.
- Z. Si, Y. He, and K. Wang, “A defect-correction method for unsteady conduction convection problems I: spatial discretization,” Science China Mathematics, vol. 54, no. 1, pp. 185–204, 2011.
- Y. Yang, W. Jiang, Y. Zhang, W. J. Wang, and H. Bi, “A two-scale discretization scheme for mixed variational formulation of eigenvalue problems,” Abstract and Applied Analysis, vol. 2012, Article ID 812914, 29 pages, 2012.
- Y. Yang, Y. Zhang, and H. Bi, “Multigrid discretization and iterative algorithm for mixed variational formulation of the eigenvalue problem of electric field,” Abstract and Applied Analysis, vol. 2012, Article ID 190768, 25 pages, 2012.
- H. Bi and Y. Yang, “Multiscale discretization scheme based on the Rayleigh quotient iterative method for the Steklov eigenvalue problem,” Mathematical Problems in Engineering, vol. 2012, Article ID 487207, 18 pages, 2012.
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