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 iterativemethod and spectralmethod to establish an efficient scheme. Finally, numerical experimentswithMATLAB program are reported.


Introduction
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 [1] and the references therein), in the analysis of the antiplane shearing on a system of collinear faults under slip-dependent friction law (see [2]), in the study of surface waves (see [3]), in the study of the vibration modes of a structure in contact with an incompressible fluid (see [4]) and vibration of a pendulum (see [5]), and in the eigenoscillations of mechanical systems with boundary conditions containing frequency [6], and thus, numerical methods for solving Steklov eigenvalue problems have received increasing attention in recent years.Bramble and Osborn [7] studied the Galerkin approximation of a Steklov eigenvalue problem of nonself-adjoint second order elliptic operators in smooth domain, Andreev and Todorov [8] 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 [9] introduced and analyzed the conforming linear finite element approximation of the Steklov eigenvalue problem in a bounded polygonal domain, Alonso and Russo [10], Yang et al. [11], and Li et al. [12] studied nonconforming finite elements approximation of the Steklov eigenvalue problem, Li and Yang [13] and Bi and Yang [14] discussed a two-grid method of the conforming and non-conforming finite element method, respectively, Li et al. [15] studied the extrapolation and superconvergence of the Steklov eigenvalue problem, Tang et al. [16] studied the boundary element approximation, and Cao et al. [17] 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 [20]) to give a priori error estimates.Based on the work of [21], we discuss a posterior error estimator of the residual type.Moreover, inspired by the work of [22] 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.

Preliminaries
We consider the following boundary value problem: where Ω ⊂  2 is a rectangular domain and / is the outward normal derivative on Ω.We denote the Sobolev space with norm ‖ ⋅ ‖  and seminorm | ⋅ |  by   (Ω).Let ‖ ⋅ ‖ 0 and ‖ ⋅ ‖ 0,Ω be the norms in the space  2 (Ω) and  2 (Ω), 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  ∈  1 (Ω), such that where Obviously, the bilinear form (⋅, ⋅) is continuous and  1 (Ω)elliptic, that is, there exists constants  and  independent of , V, such that From the Lax-Milgram theorem we know that there exists a unique solution to (2).
Find   ∈   (Ω), such that For  ≥ 3/2 and each integer  ≥ 1, we define the interpolation operator with the LGL interpolation nodes in Ω.We quote from [18] the following interpolation estimates for spectral method at the tensor-product LGL points Let  ∈   (Ω) and   be the solution of ( 2) and ( 8), respectively.Then we derive from Céa Lemma that Notice that ( 11) is also valid for special method with modal basis (see [18,19]).Using Aubin-Nitsche technique, we deduce the a priori error estimate from ( 5) and ( 10)- (11):

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 Ω ⊂  2 is a rectangular domain.
The variational problem associated with ( 13) is given by the following.
The spectral-Galerkin approximation to ( 14) is as the follows.

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   (Ω) approximating (),   () and   characterize the degrees of   (Ω) approximating the generalized solution  of (2).
Proof.We deduce from the interpolation estimate (10) that Using ( 10) and ( 6) we have From ( 10) and regularity estimate (5) we get The proof is completed.

A Posterior Error Estimates for the Steklov Eigenvalue
Problem.Theorem 2.1 in [24] gave the following results.
Remark 7. Because there has an affine transformation from rectangular domain to the unit square, Lemma 5 is suited to rectangular domain, and Lemmas 4 and 6 are valid for all edges of rectangular.
Define the global error indicator   ,  ∈ [0, 1], which is expressed as the sum of two terms: where the first term where the weight functions Φ Ω and Φ  are scaled transformations of the weight functions Φ  and Φ ê, respectively.By using the proof of [21], we can get the following results. Since The proof is completed.
Combining Lemmas 9 and 10, we obtain the following theorem.

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][26][27][28][29][30][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).
In the tables in this section, DOF denotes the degree of freedom (number of nodes).
From Table 1 we can see that We also use the bilinear finite element to solve (13), and the numerical results are listed in Table 2.
From Table 2   be the th approximate eigenvalue ( = 1, 2, 4), and the numerical results are presented in Table 3.
From Table 3 we find the eigenvalues obtained by Scheme 13 have the same accuracy with  , 2 which is computed directly in   2 (Ω) by LGL spectral method.
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 think these approximate eigenvalues have high accuracy.

Table 1 :
The eigenvalues of LGL spectral method.

Table 2 :
The eigenvalues of the bilinear finite element.

Table 3 :
The eigenvalues of shifted-inverse iterative method.