A Class of Spectral Element Methods and Its A Priori / A Posteriori Error Estimates for 2 nd-Order Elliptic Eigenvalue Problems

and Applied Analysis 3 solutions of (10) and (11), respectively; then we derive from Céa lemma and the interpolation estimates that 󵄩󵄩󵄩󵄩wN − w 󵄩󵄩󵄩󵄩1,Ω ≤ C (m)N −m+1 ‖w‖m,Ω, (18) 󵄩󵄩󵄩󵄩wN,h − w 󵄩󵄩󵄩󵄩1,Ω ≤ {∑ κ C (sκ) h 2{min(N κ +1,s κ )−1} κ × N 2(−s κ +1) κ ‖w‖ 2 s κ ,κ } 1/2


Introduction
As we know, finite element methods are local numerical methods for partial differential equations and particularly well suitable for problems in complex geometries, whereas spectral methods can provide a superior accuracy, at the expense of domain flexibility.Spectral element methods combine the advantages of the above methods (see [1]).So far, spectral and spectral element methods are widely applied to boundary value problems (see [1,2]), as well as applied to symmetric eigenvalue problems (see [3]).However, it is still a new subject to apply them to nonsymmetric elliptic eigenvalue problems.
Based on the work mentioned above, this paper shall further apply spectral and spectral element methods to nonsymmetric elliptic eigenvalue problems.This paper will mainly perform the following work.
(1) We prove a priori and a posteriori error estimates of spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis, respectively, for the general 2nd-order elliptic eigenvalue problems.
(2) We compare between spectral methods, spectral element methods with Legendre-Gauss-Lobatto nodal basis, finite element methods, and their derived version, ℎ-version, and ℎ-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods for nonsymmetric 2nd-order elliptic eigenvalue problems.
This paper is organized as follows.Section 2 introduces basic knowledge of second elliptic eigenvalue problems.Sections 3 and 4 are devoted to a priori and a posteriori error estimates of spectral and spectral element methods, respectively.In Section 5, some numerical experiments are performed by the methods mentioned above.
In this paper,  denotes a generic positive constant independent of the polynomial degrees and mesh scales, which may not be the same at different occurrences.
We assume that  ℎ = {} is a regular rectangle (resp.cuboid) or simplex partition of the domain Ω and satisfies Ω = ⋃ .We associate with the partition a polynomial degree vector N = {  }, where   is the polynomial degree in .Let ℎ  be the diameter of the element , and let ℎ = max ∈ ℎ ℎ  .
We define spectral and spectral element spaces as follows: where   (Ω) and    () are polynomial spaces of degree  (resp.degree  in every direction) in Ω and degree   (resp.degree   in every direction) in the element , respectively.The spectral approximation of ( 5) is as follows: find   ∈   (Ω), such that The spectral element approximation of ( 5) is as follows: find  ,ℎ ∈  ,ℎ (Ω), such that We assume that  ∈  2 (Ω) and derive from Lax-Milgram theorem that the variational formations ( 5), ( 6), (10), and ( 11) have a unique solution, respectively.
Define the interpolation operators as the interpolations in the element  and the domain Ω, respectively, with the tensorial Legendre-Gauss-Lobatto (LGL) interpolation nodes.Define the interpolation operator We quote from [2] (see (5.8.27) therein) the interpolation estimates for spectral and spectral element methods with LGL Nodal-basis as follows.
Using Aubin-Nitsche technique, we deduce from the regularity estimate (8) and the estimates ( 18)-( 20) the priori estimates of boundary value problem (5) for spectral and spectral element methods; that is,

A Priori Error
Estimates.We will analyze a prior error estimates for spectral element methods which are suitable for spectral methods with mesh fineness ℎ not considered.

A Posteriori Error Estimates
Based on [20], we will discuss a posteriori error estimates.We further assume that Ω ⊂  2 , the partition  ℎ is -shape regular, and the polynomial degree of neighboring elements are comparable; that is, there exists  > 0, such that for all ,   ∈  ℎ ,  ∩   ̸ = 0, We refer to the ℎ-clément interpolation estimates given by [20,21] (see theorems 2.2 and 2.3, respectively), which generalize the well-known clément type interpolation operators studied in [22] and [23] to the hp context.Lemma 8. Assume that the partition  ℎ is -shape regular and the polynomial distribution N is comparable.Then there exists a positive constant  = () and the clément operator  :  1 0 (Ω) →  ,ℎ (Ω), such that where ℎ  is the length of the edge  and   = max(  1 ,   2 ), where  1 ,  2 are elements sharing the edge  and   ,   are patches covering  and  with a few layers, respectively.Define interval Î = (0, 1) and weight function Φ Î() := (1 − ).Denote the reference square and triangle element by κ = (0, 1) The following three lemmas are given by [20].Lemmas 9-10 provide the polynomial inverse estimates in standard interval and element, while Lemma 11 provides a result for the extension from an edge to the element.
Then there exists  = (, ), such that for all  ∈ N and all univariate polynomials   of degree , (61) Lemma 10.Let −1 <  < ,  ∈ [0, 1].Then there exist  1 = (, ),  2 =   > 0, such that for all  ∈ N and all polynomials   of degree bi-, then there exists   > 0 such that for all  ∈ N,  ∈ (0, 1], and all univariate polynomials  of degree , there exists an extension V ê ∈  1 (κ) and holds It is easy to know that the three lemmas above hold for complex-valued polynomials.
Combining Lemmas 14 and 15, we obtain the following theorem.

Numerical Experiments
In this section, we simply denote spectral methods, spectral element methods, and finite element methods with SM, SEM, and FEM, respectively.And spectral methods with equidistant nodal basis, modal basis, and LGL nodal basis are replaced by Eq-SM, Modal-SM, and LGL-SM, respectively.Note that all these methods employ the tensorial basis.

Example 1. Consider the nonsymmetric eigenvalue problem
The first eigenvalue of (109)  1 = 101/4 + 2 2 is a simple eigenvalue.And the corresponding eigenfunctions are sufficiently smooth.

Comparisons between LGL-SM, Modal, and Eq-SM.
Figure 1 shows that the condition numbers of the first eigenvalue for LGL-SM, Modal-SM, and Eq-SM coincide with each other at the beginning but perform abnormally with  > 19 for Eq-SM.Table 1 tells us that when  > 11, the accuracy of first eigenvalue obtained by Eq-SM is not as good as obtained that by LGL-SM and Modal-SM.When  = 15, the error of the first eigenvalues obtained by Eq-SM is greater than 1E-5; however, the order of the magnitude of errors for LGL-SM and Modal-SM still keeps below 1E-13.The best result of first eigenvalue error for Eq-SM is merely 1E-9 or so.

5.1.2.
LGL-SM and Modal-SM versus hp-SEM.Tables 1 and  2 indicate that increasing the polynomial degree  or  decreasing the mesh fineness h can decrease the errors of the first eigenvalue.But it is expensive to increase polynomial degree and decrease mesh fineness h at the same time.For ℎ = 1/4 and ℎ = 1/16, we obtain from Table 2 the first eigenvalue errors 2.8 − 14 and 1.3 − 13 and the corresponding degree of freedom 1225 and 6241 for hp-SEM, respectively, Whereas from Table 1, to reach this accuracy, LGL-SM and Modal-SM should merely perform the interpolation approximations with polynomial degree bi-14 and bi-13 or so, and the corresponding degrees of freedom are merely 169 and 144, respectively.Therefore, we conclude that LGL-SM and Modal-SM are highly accurate and efficient for this kind of nonsymmetric eigenvalue problems.
In Figure 2 from [9], when the degree of freedom is up to 1000, the error of linear FEM is about 1E-2; the function value recovery techniques in [9] obviously improves the accuracy up to 1E-5.Comparing Tables 1 and 2 in this paper with Figure 2 in [9], we can also find the advantages of LGL-SM, Modal-SM, and hp-SEM over the function value recovery techniques for FEM given by [9] from accuracy and degree of freedom.

hp-SEM versus hp-FEM.
From Table 4, we find that the condition number of the first eigenvalue for hp-version methods (hp-SEM and hp-FEM) stays at 4.27.It is indicated from Tables 2 and 3 that, when  is greater than 7, compared with hp-SEM, the errors of hp-FEM tend to become large, whereas the errors of hp-SEM still keep stable or even stay a decreasing tendency; however, this phenomenon is not apparent for ℎ = 1/2.Remark 21.Condition numbers of 1st eigenvalue for hp-FEM (not listed in Table 4) are almost the same to those for hp-SEM.
From Theorem 19, we know that   is a reliable error indicator for  ,ℎ .We choose  0 (setting  = 0 in (110)) as a posteriorii error indicator.
In Figures 2 and 3, we denote the true error and est.error with | ,ℎ − | and  0 , respectively.
As is depicted in Figure 2, when the polynomial degree  ≤ 12, the error indicator  0 can properly estimate the true errors of LGL-SM for the first eigenvalue, however, also slightly underestimate the true errors.It is easy to see that  0 shows almost the same algebraic decay as the true error with the polynomial degree  (≤12) increasing.Nevertheless, the error indicator  0 cannot approximate the true errors if  is large enough, which is caused by round-off errors derived from the bad condition number of eigenvalue.In Figure 3, we give the comparison between the error indicator  0 and the true errors for hp-SEM.

Example 2. Consider the nonsymmetric eigenvalue problem
−Δ + 10  = , in Ω = (−1, 1) 2 (0, 1) 2 ,  = 0, on Ω. (111) A reference value for the first eigenvalue (simple eigenvalue) of (111) is 34.6397 given by [5].And the corresponding eigenfunctions have the singularity at the origin.Next, we shall compare the relevant numerical results between P-SEM and the other methods adopted in this paper.Note that here and hereafter P-version methods are for the fixed mesh fineness ℎ = 1.Table 5 lists part data of the approximate eigenvalues computed by P-SEM and the corresponding error indicator  0 for reference.

Stability of P-Version
Methods. Figure 4 indicates that the eigenvalues computed by P-FEM will not seriously deviate from the results computed by P-SEM until the interpolation polynomial degree  is up to 19.This phenomenon coincides with the abnormity of condition number of first eigenvalue for P-FEM (see Figure 5).The reason is that the singularities of the eigenfunctions limit the accuracy of both kinds of methods; this is slightly different from the case of the eigenvalue problem with the sufficiently smooth eigenfunctions.

P-SEM versus Other Methods
. By calculations, we find that, in the case of the linear FEM, for fixed mesh fineness ℎ = 1/256, the approximate eigenvalue is 34.6403 with degree of freedom up to 195585.But P-SEM with the polynomial degree bi-22 can reach this accuracy, and the corresponding   degree of freedom is merely 1365.Compared with the linear FEM, hp-SEM can obtain a higher accuracy with less degrees of freedom as follows: for fixed ℎ = 1/16 and  = 10, the approximate eigenvalue is 34.63984 with degree of freedom 76161 but P-SEM with polynomial degree bi-44 can reach this accuracy.Therefore, P-SEM is more efficient for the eigenvalue problems with the singular solutions than the other methods.

Figure 1 :
Figure 1: Condition number of first eigenvalue for SM.

Figure 4 :
Figure 4: The Approximate 1st eigenvalue of P-SEM and P-FEM.

Figure 5 :
Figure 5: Condition number of first eigenvalue for P-SEM and P-FEM.

Table 2 :
Errors and DOF of hp-SEM for the first eigenvalue.

Table 3 :
Errors of hp-FEM for the first eigenvalue.

Table 4 :
Condition number of first eigenvalue for hp-SEM.

Table 5 :
The Approximate eigenvalues and indicator  0 of P-SEM.