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

Abstract

This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and their derived 𝑝-version, ℎ-version, and ℎ𝑝-version methods from accuracy, degree of freedom, and stability and verify that spectral methods and spectral element methods are highly efficient computational methods.

1. 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.

A posteriorii error estimates and highly efficient computational methods for finite elements of eigenvalue problems are the subjects focused on by the academia these years; see [3–16], and among them, for nonsymmetric 2nd-order elliptic eigenvalue problems, [5, 15] provide a posteriori error estimates and adaptive algorithms, [9] the function value recovery techniques and [8, 10] two-level discretization schemes.

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 𝑝-version, ℎ-version, and ℎ𝑝-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.

2. Preliminaries

Consider the 2nd-order elliptic boundary value problem 𝐿𝑤=−∇⋅(𝐷∇𝑤)+𝐛⋅∇𝑤+𝑐𝑤=𝑓,inΩ,𝑤=0,on𝜕Ω, (1) where Ω⊂𝑅^𝑑(𝑑=2,3) is a bounded domain, 𝐛 and 𝑐 are a real-valued vector function and a real-valued function, respectively, and 𝐷 is a positive scalar function with 𝐷(𝑥)≥𝐷₀>0(∀𝑥∈Ω).

We denote the complex Sobolev spaces with norm ‖⋅‖_𝑚 by 𝐻^𝑚(Ω), 𝐻¹₀(Ω)={𝑣∈𝐻¹(Ω),𝑣∣_𝜕Ω=0}. Let (⋅,⋅) and ‖⋅‖_0,Ω be a inner product and a norm in the complex space 𝐿²(Ω), respectively.

In this paper, 𝐶 denotes a generic positive constant independent of the polynomial degrees and mesh scales, which may not be the same at different occurrences.

Define the bilinear form 𝑎(⋅,⋅) as follows: 𝑎(𝑤,𝑣)=_Ω𝐷∇𝑤∇𝑣+(𝐛⋅∇𝑤)𝑣+𝑐𝑤𝑣,∀𝑤,𝑣∈𝐻¹₀(Ω). (2)

We assume that 𝑓∈𝐿²(Ω), 𝐷,𝐛, and 𝑐 are bounded functions on Ω, namely 𝐷,𝑐∈𝐿^∞(Ω),𝐛∈(𝐿^∞(Ω))^𝑑. Further more, we assume that ∇⋅𝐛 exists and satisfies −12∇⋅𝐛+𝑐≥0,inΩ. (3)

Under these assumptions, the bilinear form 𝑎(⋅,⋅) is continuous in 𝐻¹₀(Ω) and 𝐻¹₀(Ω)-elliptic; that is, there exist two constants 𝐴,𝐵>0 independent of 𝑤,𝑣 such that |𝑎(𝑤,𝑣)|≤𝐴‖𝑤‖_1,Ω‖𝑣‖_1,Ω,∀𝑤,𝑣∈𝐻¹₀(Ω),Re𝑎(𝑣,𝑣)≥𝐵‖𝑣‖²_1,Ω,∀𝑣∈𝐻¹₀(Ω). (4)

The corresponding variational formulation of (1) is given as follows: find 𝑤∈𝐻¹₀(Ω), such that 𝑎(𝑤,𝑣)=𝑓,𝑣,∀𝑣∈𝐻¹₀(Ω). (5)

The adjoint problem of (5) is as follows: find 𝑤^∗∈𝐻¹₀(Ω), such that 𝑎𝑣,𝑤^∗=𝑣,𝑓,∀𝑣∈𝐻¹₀(Ω). (6)

As the general 2nd-order elliptic boundary value problems, we assume that the regularity estimates for problem (5) and its adjoint problem (6) hold, respectively. Namely ‖𝑤‖_𝑟1+1,Ω≤𝐶𝑓_0,Ω,0<𝑟₁𝑤≤1,_{∗𝑟2+1,Ω}≤𝐶𝑓_0,Ω,0<𝑟₂≤1. (7)(8)

We assume that 𝐾_ℎ={𝜅} is a regular rectangle (resp. cuboid) or simplex partition of the domain Ω and satisfies Ω=⋃𝜅. We associate with the partition a polynomial degree vector 𝐍={𝑁_𝜅}, where 𝑁_𝜅 is the polynomial degree in 𝜅. Let ℎ_𝜅 be the diameter of the element 𝜅, and let ℎ=max_{𝜅∈𝐾ℎ}ℎ_𝜅.

We define spectral and spectral element spaces as follows: 𝑆_𝑁(Ω)=𝑣∈𝑃_𝑁(Ω),𝑣|_𝜕Ω,𝑆=0_𝑁,ℎ(Ω)=𝑣∈𝐶Ω∶𝑣|_𝜅∈𝑃_𝑁𝜅(𝜅),∀𝜅∈𝐾_ℎ,𝑣|_𝜕Ω=0, (9) where 𝑃_𝑁(Ω) and 𝑃_𝑁𝜅(𝜅) are polynomial spaces of degree 𝑁 (resp. degree 𝑁 in every direction) in Ω and degree 𝑁_𝜅 (resp. degree 𝑁_𝜅 in every direction) in the element 𝜅, respectively.

The spectral approximation of (5) is as follows: find 𝑤_𝑁∈𝑆_𝑁(Ω), such that 𝑎𝑤_𝑁,𝑣=𝑓,𝑣,∀𝑣∈𝑆_𝑁(Ω). (10)

The spectral element approximation of (5) is as follows: find 𝑤_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω), such that 𝑎𝑤_𝑁,ℎ,𝑣=𝑓,𝑣,∀𝑣∈𝑆_𝑁,ℎ(Ω). (11)

We assume that 𝑓∈𝐿²(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 𝐼_{𝑁𝜅,ℎ𝜅}∶𝐶(𝜅)⟶𝑃_𝑁𝜅(𝜅),𝐼_𝑁∶𝐶(Ω)⟶𝑆_𝑁(Ω), (12) as the interpolations in the element 𝜅 and the domain Ω, respectively, with the tensorial Legendre-Gauss-Lobatto (LGL) interpolation nodes.

Define the interpolation operator 𝐼_𝑁,ℎ∶𝐶(Ω)⟶𝑆_𝑁,ℎ(Ω),satisfying𝐼_𝑁,ℎ|_𝜅=𝐼_{𝑁𝜅,ℎ𝜅}. (13)

We quote from [2] (see (5.8.27) therein) the interpolation estimates for spectral and spectral element methods with LGL Nodal-basis as follows.

For all 𝑣∈𝐻^𝑠𝜅(𝜅),𝑠_𝜅≥(𝑑+1)/2, 𝑣−𝐼_{𝑁𝜅,ℎ𝜅}𝑣_1,𝜅≤𝐶𝑠_𝜅ℎ^{min(𝑁𝜅+1,𝑠𝜅}_𝜅⁾⁻¹𝑁^−𝑠𝜅_𝜅⁺¹‖𝑣‖_{𝑠𝜅,𝜅},𝑣−𝐼_{𝑁𝜅,ℎ𝜅}𝑣_0,𝜅≤𝐶𝑠_𝜅ℎ^{min(𝑁𝜅+1,𝑠𝜅)}_𝜅𝑁^−𝑠𝜅_𝜅‖𝑣‖_{𝑠𝜅,𝜅}. (14)(15)

For all 𝑣∈𝐻^𝑠(Ω),𝑠≥(𝑑+1)/2, 𝑣−𝐼_𝑁𝑣_1,Ω≤𝐶(𝑠)𝑁^−𝑠+1‖𝑣‖_𝑠,Ω,𝑣−𝐼_𝑁𝑣_0,Ω≤𝐶(𝑠)𝑁^−𝑠‖𝑣‖_𝑠,Ω. (16)(17)

We assume that the solution of boundary value problem (5) 𝑤∈𝐻¹₀(Ω)∩𝐻^𝑚(Ω)(𝑚>1), that 𝑤_𝑁 and 𝑤_𝑁,ℎ are the solutions of (10) and (11), respectively; then we derive from Céa lemma and the interpolation estimates that 𝑤_𝑁−𝑤_1,Ω≤𝐶(𝑚)𝑁^−𝑚+1‖𝑤‖_𝑚,Ω,𝑤_𝑁,ℎ−𝑤_1,Ω≤_𝜅𝐶𝑠_𝜅ℎ^{2{min(𝑁𝜅+1,𝑠𝜅}_𝜅^)−1}×𝑁^{2(−𝑠𝜅}_𝜅⁺¹⁾‖𝑤‖²_{𝑠𝜅,𝜅}^1/2, (18)(19) where 𝑠_𝜅≥(𝑑+1)/2,∀𝜅∈𝐾_ℎ.

Particularly, if 𝑁_𝜅=𝑁,∀𝜅∈𝐾_ℎ, then we have 𝑤_𝑁,ℎ−𝑤_1,Ω≤𝐶(𝑚)ℎ^{min(𝑁+1,𝑚)−1}×𝑁^1−𝑚‖𝑤‖_𝑚,Ω. (20)

Note that (18) is also suited to spectral methods with modal basis (see [1, 2]).

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, 𝑤_𝑁−𝑤_0,Ω≤𝐶𝑁^{−𝑚−𝑟2+1}‖𝑤‖_𝑚,Ω,𝑤_𝑁,ℎ−𝑤_0,Ω≤𝐶𝑁^{−𝑚−𝑟2+1}ℎ^{𝑟2+min(𝑁+1,𝑚)−1}‖𝑤‖_𝑚,Ω. (21)(22)

3. Spectral and Spectral-Element Approximations and Error Estimates for Eigenvalue Problems

3.1. Spectral and Spectral-Element Approximations for Eigenvalue Problems

Consider the following eigenvalue problem corresponding to the boundary value problem (1): 𝐿𝑢=𝜆𝑢,inΩ,𝑢=0,on𝜕Ω. (23)

The variational formation of (23) is given by the following: find 𝜆∈ℂ,0≠𝑢∈𝐻¹₀(Ω), such that 𝑎(𝑢,𝑣)=𝜆(𝑢,𝑣),∀𝑣∈𝐻¹₀(Ω). (24)

The spectral approximation scheme of (24) is given by the following: find 𝜆_𝑁∈ℂ,0≠𝑢_𝑁∈𝑆_𝑁(Ω), such that 𝑎𝑢_𝑁,𝑣_𝑁=𝜆_𝑁𝑢_𝑁,𝑣_𝑁,∀𝑣_𝑁∈𝑆_𝑁(Ω). (25)

The spectral element approximation scheme of (24) is given by the following: find 𝜆_𝑁,ℎ∈ℂ,0≠𝑢_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω), such that 𝑎𝑢_𝑁,ℎ,𝑣_𝑁,ℎ=𝜆_𝑁,ℎ𝑢_𝑁,ℎ,𝑣_𝑁,ℎ,∀𝑣_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω), (26)

Define the solution operators 𝑇∶𝐿²(Ω)→𝐻¹₀(Ω),𝑇_𝑁∶𝐿²(Ω)→𝑆_𝑁(Ω), and 𝑇_𝑁,ℎ∶𝐿²(Ω)→𝑆_𝑁,ℎ(Ω) by 𝑎𝑇𝑓,𝑣=𝑓,𝑣,∀𝑓∈𝐿²(Ω),∀𝑣∈𝐻¹₀(Ω),𝑎𝑇_𝑁𝑓,𝑣_𝑁=𝑓,𝑣_𝑁,∀𝑓∈𝐿²(Ω),∀𝑣_𝑁∈𝑆_𝑁(Ω),𝑎𝑇_𝑁,ℎ𝑓,𝑣_𝑁,ℎ=𝑓,𝑣_𝑁,ℎ,∀𝑓∈𝐿²(Ω),∀𝑣_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω). (27) Obviously (see [17]), the equivalent operator forms for (24) and (26) are the following.

Find 𝜆∈ℂ,0≠𝑢∈𝐻¹₀(Ω), such that 𝑇𝑢=𝜆⁻¹𝑢. (28)

Find 𝜆_𝑁,ℎ∈ℂ,0≠𝑢_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω), such that 𝑇_𝑁,ℎ𝑢_𝑁,ℎ=𝜆⁻¹_𝑁,ℎ𝑢_𝑁,ℎ. (29) The adjoint problem of the eigenvalue problem (23) is 𝐿^∗𝑢^∗=𝜆^∗𝑢^∗𝑢,inΩ,_∗=0,on𝜕Ω, (30) where 𝐿^∗𝑢^∗=−∇⋅(𝐷∇𝑢^∗)−𝐛⋅∇𝑢^∗+(𝑐−∇⋅𝐛)𝑢^∗.

The variational formation of (30) is given by the following: find 𝜆^∗∈ℂ,0≠𝑢^∗∈𝐻¹₀(Ω), such that 𝑎^∗𝑢^∗,𝑣∶=𝑎(𝑣,𝑢^∗)=𝜆^∗𝑢^∗,𝑣,∀𝑣∈𝐻¹₀(Ω). (31)

The spectral element approximation scheme of (31) is given by the following: find 𝜆^∗_𝑁,ℎ∈ℂ,0≠𝑢^∗_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω), such that 𝑎^∗𝑢^∗_𝑁,ℎ,𝑣_𝑁,ℎ=𝜆^∗_𝑁,ℎ𝑢^∗_𝑁,ℎ,𝑣_𝑁,ℎ,∀𝑣_𝑁,ℎ∈𝑆_𝑁,ℎ(Ω). (32)

We can likewise define the equivalent operator forms for the eigenvalue problems (31) and (32) as 𝑇^∗𝑢^∗=𝜆^∗−1𝑢^∗,𝑇^∗_𝑁,ℎ𝑢^∗_𝑁,ℎ=𝜆^∗−1_𝑁,ℎ𝑢^∗_𝑁,ℎ. (33)

Let 𝜆 be an eigenvalue of (23). There exists a smallest integer 𝜇, called the ascent of 𝜆, such that ker((𝜆⁻¹−𝑇)^𝜇)=ker((𝜆⁻¹−𝑇)^𝜇+1). 𝑞=dimker((𝜆⁻¹−𝑇)^𝜇) is called the algebraic multiplicity of 𝜆. The functions in ker((𝜆⁻¹−𝑇)^𝜇) are called generalized eigenfunctions of 𝑇 corresponding to 𝜆. Likewise the ascent, algebraic multiplicity and generalized eigenfunctions of 𝜆_𝑁,ℎ, 𝜆^∗ and 𝜆^∗_𝑁,ℎ can be defined.

Let 𝜆 be an eigenvalue of (23) with the algebraic multiplicity 𝑞 and the ascent 𝜇. Assume ‖𝑇_𝑁,ℎ−𝑇‖_1,Ω→0(𝑁→∞,ℎ→0); then there are eigenvalues 𝜆_{𝑗,𝑁,ℎ} (𝑗=1,2,…,𝑞) of (26) which converge to 𝜆. Let 𝑀(𝜆) be the space spanned by all generalized eigenfunctions corresponding to 𝜆 of 𝑇, and let 𝑀_𝑁,ℎ(𝜆) be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of 𝑇_𝑁,ℎ that converge to 𝜆.

In view of adjoint problems (31) and (32), the definitions of 𝑀^∗(𝜆^∗) and 𝑀^∗_𝑁,ℎ(𝜆^∗) are analogous to 𝑀(𝜆) and 𝑀_𝑁,ℎ(𝜆). Let 𝑀(𝜆)={𝑣∈𝑀(𝜆)∶‖𝑣‖_1,Ω=1}, and let 𝑀^∗(𝜆^∗)={𝑣∈𝑀^∗(𝜆^∗)∶‖𝑣‖_1,Ω=1}.

Note that when 𝐛=0, both (24) and (26) are symmetric. Thus, the ascent 𝜇=1 of 𝜆, and the ascent 𝑙=1 of 𝜆_𝑁,ℎ.

3.2. 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.

Assume that 𝑅 and 𝑈 are two closed subspace in 𝐻¹₀(Ω).

Denote 𝛿(𝑅,𝑈)=sup_{𝑣∈𝑅‖𝑣‖1,Ω=1}(),𝜃()(𝛿()(.dist𝑣,𝑈𝑅,𝑈=max𝑅,𝑈,𝛿𝑈,𝑅)) (34) We say that 𝜃(𝑅,𝑈) is the gap between 𝑅 and 𝑈.

Denote 𝜀_𝑁,ℎ=𝜀_𝑁,ℎ(𝜆)=sup_𝑢∈𝑀_(𝜆)inf_{𝑣∈𝑆𝑁,ℎ(Ω)}‖𝑢−𝑣‖_1,Ω,𝜀^∗_𝑁,ℎ=𝜀^∗_𝑁,ℎ𝜆^∗=sup_𝑢∈𝑀^∗_(𝜆^∗₎inf_{𝑣∈𝑆𝑁,ℎ(Ω)}‖𝑢−𝑣‖_1,Ω. (35)

We give the following four lemmas from Theorem 8.1–8.4 in [17], which are applications to spectral element methods.

Lemma 1. Assume ‖𝑇_𝑁,ℎ−𝑇‖_1,Ω→0(𝑁→∞,ℎ→0). For small enough ℎ and big enough 𝑁, there holds 𝜃𝑀(𝜆),M_𝑁,ℎ(𝜆)≤𝐶𝜀_𝑁,ℎ. (36)

Lemma 2. Assume ‖𝑇_𝑁,ℎ−𝑇‖_1,Ω→0(𝑁→∞,ℎ→0); then 𝜆⁻¹−1𝑞^𝑞_𝑗=1𝜆⁻¹_{𝑗,𝑁,ℎ}≤𝐶𝜀_𝑁,ℎ𝜀^∗_𝑁,ℎ. (37)

Lemma 3. Assume that ‖𝑇_𝑁,ℎ−𝑇‖_1,Ω→0(𝑁→∞,ℎ→0); then there holds 𝜆−𝜆_{𝑗,𝑁,ℎ}≤𝐶𝜀_𝑁,ℎ𝜀^∗_𝑁,ℎ^1/𝜇𝑗=1,2,…,𝑞. (38) Since ker((𝜆⁻¹−𝑇)^𝑙)(𝑙≥1) is a finite dimensional space, there exists a direct-sum decomposition 𝐻¹₀(Ω)=ker((𝜆⁻¹−𝑇)^𝑙)⊕𝑀_𝑙. We define the operator 𝐸_𝑙 as a projection along 𝑀_𝑙 from 𝐻¹₀(Ω) to ker((𝜆⁻¹−𝑇)^𝑙).

Lemma 4. Assume ‖𝑇_𝑁,ℎ−𝑇‖_1,Ω→0(𝑁→∞,ℎ→0). Let 𝜆_𝑁,ℎ be an eigenvalue of 𝑇_𝑁,ℎ and lim_{𝑁→∞,ℎ→0}𝜆_𝑁,ℎ=𝜆. 𝑢_𝑁,ℎ satisfies (𝜆⁻¹_𝑁,ℎ−𝑇_𝑁,ℎ)^𝑘𝑢_𝑁,ℎ=0 and ‖𝑢_𝑁,ℎ‖_1,Ω=1, where 𝑘≤𝜇 is a positive integer. Then, for every integer 𝑙∈[𝑘,𝜇], there holds 𝑢_𝑁,ℎ−𝐸_𝑙𝑢_𝑁,ℎ1,Ω≤𝐶𝜀^{(𝑙−𝑘+1)/𝜇}_𝑁,ℎ. (39)
We assume that in this section, for the sake of simplicity, 𝑁_𝜅=𝑁,∀𝜅∈𝐾_ℎ.

Theorem 5. If 𝑀(𝜆)⊂𝐻^𝑡1(Ω) and 𝑀^∗(𝜆^∗)⊂𝐻^𝑡2(Ω), then there holds the following error estimates: 1𝑞^𝑞_𝑗=1𝜆_{𝑗,𝑁,ℎ}ℎ−𝜆≤𝐶^{𝜏1+𝜏2−2}𝑁^{𝑡1+𝑡2−2}sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ωsup_𝑣∈𝑀^∗_𝜆^∗‖𝑣‖_𝑡2,Ω,𝜆_{𝑗,𝑁,ℎ}ℎ−𝜆≤𝐶^{𝜏1+𝜏2−2}𝑁^{𝑡1+𝑡2−2}sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ωsup_𝑣∈𝑀^∗_𝜆^∗‖𝑣‖_𝑡2,Ω^1/𝜇,𝜃𝑀(𝜆)𝑗=1,2,…,𝑞,𝑀_𝑁,ℎ(𝜆)ℎ≤𝐶^𝜏1−1𝑁^𝑡1−1sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ω (40)(41)(42) Let ‖𝑢_𝑁,ℎ‖_1,Ω=1, and let (𝜆⁻¹_𝑁,ℎ−𝑇_𝑁,ℎ)^𝑙1𝑢_𝑁,ℎ=0, for some 𝑙₁≤𝜇. Then, for every integer 𝑙₂(𝑙₁≤𝑙₂≤𝜇), there exists a function 𝑢^′, such that (𝜆⁻¹−𝑇)^𝑙2𝑢^′=0 and 𝑢_𝑁,ℎ−𝑢^′_1,Ωℎ≤𝐶^𝜏1−1𝑁^𝑡1−1sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ω^{(𝑙2−𝑙1+1)/𝜇}, (43) where 𝜏₁=min(𝑁+1,𝑡₁), 𝜏₂=min(𝑁+1,𝑡₂).

Proof. We derive from the error estimate (20) that 𝑇_𝑁,ℎ−𝑇_1,Ω=sup_{𝑓∈𝐻10(Ω)}𝑇−𝑇_𝑁,ℎ𝑓_1,Ω𝑓_1,Ω≤𝐶1+𝑟₁ℎ^𝑟1𝑁^−𝑟1⟶0(𝑁⟶∞,ℎ⟶0). (44) By (14), 𝜀_𝑁,ℎ=𝜀_𝑁,ℎ(𝜆)=sup_𝑢∈𝑀_(𝜆)inf_{𝑣∈𝑆𝑁,ℎ(Ω)}‖𝑢−𝑣‖_1,Ωℎ≤𝐶^𝜏1−1𝑁^𝑡1−1sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ω. (45) Analogically, 𝜀^∗_𝑁,ℎℎ≤𝐶^𝜏2−1𝑁^𝑡2−1sup_𝑢∈𝑀^∗_(𝜆^∗₎‖𝑢‖_𝑡2,Ω. (46) Plugging the two inequalities above into (36), (38), and (39) yields (42), (41), and (43), respectively. We find from (37) that 1𝑞^𝑞_𝑗=1𝜆_{𝑗,𝑁,ℎ}=1−𝜆𝑞^𝑞_𝑗=1𝜆⁻¹_{𝑗,𝑁,ℎ}−𝜆⁻¹𝜆⁻¹𝜆⁻¹_{𝑗,𝑁,ℎ}1≤𝐶𝑞^𝑞_𝑗=1𝜆⁻¹_{𝑗,𝑁,ℎ}−𝜆⁻¹≤𝐶𝜀_𝑁,ℎ𝜀^∗_𝑁,ℎ, (47) combining with (45) and (46) yields (40).

Supposing that ‖𝑇_𝑁,ℎ−𝑇‖_0,Ω→0(𝑁→∞,ℎ→0), 𝜌(𝑇) is a regular set of 𝑇, and Γ⊂𝜌(𝑇) is a closed Jordan curve enclosing 𝜆⁻¹.

Denote 𝑅(𝑧)=(𝑇−𝑧)⁻¹,𝑅𝑇_𝑁,ℎ,𝑧=𝑇_𝑁,ℎ−𝑧⁻¹. (48)

Define the spectral projection operators 𝐸=−12𝑖𝜋_Γ𝑅(𝑇,𝑧)dz∶𝐻¹₀(Ω)(𝜆),𝐸⟶𝑀_𝑁,ℎ=−12𝑖𝜋_Γ𝑅𝑇_𝑁,ℎ,𝑧dz∶𝐻¹₀(Ω)⟶𝑀_𝑁,ℎ(𝜆). (49)

We give the following lemma by referring to [18, 19] (see proposition 5.3 in [18] and theorem 1.3.2 in [19]).

Lemma 6. If ‖𝑇_𝑁,ℎ−𝑇‖_0,Ω→0(𝑁→∞,ℎ→0), then there holds that 𝐸_𝑁,ℎ→𝐸(𝑝), 𝑅(𝑇_𝑁,ℎ,𝑧) is uniformly bounded with 𝑁 and ℎ, and 𝐸_𝑁,ℎ−𝐸𝑣_0,Ω≤𝐶max_𝑧∈Γ𝑇−𝑇_𝑁,ℎ𝑅(𝑧)𝑣_0,Ω,∀𝑣∈𝐻¹₀(Ω),𝐸_𝑁,ℎ−𝐸𝑣_0,Ω≤𝐶max_𝑧∈Γ𝑇−𝑇_𝑁,ℎ𝑅𝑇_𝑁,ℎ,𝑧𝑣_0,Ω,∀𝑣∈𝐻¹₀(Ω). (50)

Theorem 7. Under the assumptions of Theorem 5, further assume that the ascent of 𝜆 is 𝜇=1. Let (𝜆_𝑁,ℎ,𝑢_𝑁,ℎ) be an eigenpair of (26) with ‖𝑢_𝑁,ℎ‖_0,Ω=1; then there exists an eigenpair (𝜆,𝑢) of (24), such that 𝑢_𝑁,ℎ−𝑢_1,Ω≤𝐶ℎ^𝜏1−1‖𝑢‖_𝑡1,Ω𝑁^𝑡1−1,𝑢_𝑁,ℎ−𝑢_0,Ω≤𝐶ℎ^{𝑟2+𝜏1−1}‖𝑢‖_𝑡1,Ω𝑁^{𝑟2+𝑡1−1},𝜆_𝑁,ℎℎ−𝜆≤𝐶^{𝜏1+𝜏2−2}𝑁^{𝑡1+𝑡2−2}sup_𝑢∈𝑀_(𝜆)‖𝑢‖_𝑡1,Ωsup_𝑣∈𝑀^∗_𝜆^∗‖𝑣‖_𝑡2,Ω, (51)(52)(53) where 𝜏₁=min(𝑁+1,𝑡₁) and 𝜏₂=min(𝑁+1,𝑡₂).
Let (𝜆,𝑢) be an eigenpair of (24). If 𝜆_𝑁,ℎ is an eigenvalue of (26) convergence to 𝜆, then there exists 𝑢_𝑁,ℎ∈ker(𝜆⁻¹_𝑁,ℎ−𝑇_𝑁,ℎ), such that (51)–(53) hold.

Proof. We deduce (53) immediately from (41). We derive from (22) and (7) that 𝑇𝑓−𝑇_𝑁,ℎ𝑓_0,Ω≤𝐶𝑁^{−𝑟1−𝑟2}ℎ^𝑟1+𝑟2𝑓_0,Ω; (54) thus,‖𝑇−𝑇_𝑁,ℎ‖_0,Ω→0(𝑁→∞,ℎ→0). Taking 𝑢=𝐸𝑢_𝑁,ℎ and by virtue of 𝑅(𝑇_𝑁,ℎ,𝑧)𝑢_𝑁,ℎ=(𝜆⁻¹_𝑁,ℎ−𝑧)⁻¹𝑢_𝑁,ℎ, Lemma 6 and (22), we have 𝑢−𝑢_𝑁,ℎ0,Ω=𝐸𝑢_𝑁,ℎ−𝐸_𝑁,ℎ𝑢_𝑁,ℎ0,Ω≤𝐶𝑇−𝑇_𝑁,ℎ𝑢_𝑁,ℎ0,Ω≤𝐶𝑇−𝑇_𝑁,ℎ𝑢_0,Ω+𝑇−𝑇_𝑁,ℎ𝑢_𝑁,ℎ−𝑢_0,Ω, (55) from which follows 𝑢−𝑢_𝑁,ℎ0,Ω≤𝐶𝑇−𝑇_𝑁,ℎ𝑢_0,Ω≤𝐶ℎ^{𝑟2+𝜏1−1}‖𝑢‖_𝑡1,Ω𝑁^{𝑟2+𝑡1−1}, (56) which is (52). By direct calculation, we have 𝑢−𝑢_𝑁,ℎ1,Ω=𝜆𝑇𝑢−𝜆_ℎ𝑇_𝑁,ℎ𝑢_𝑁,ℎ1,Ω≤𝜆𝑇𝑢−𝜆𝑇_𝑁,ℎ𝑢_1,Ω+𝜆𝑇_𝑁,ℎ𝑢−𝜆_ℎ𝑇_𝑁,ℎ𝑢_𝑁,ℎ1,Ω≤𝑇−𝑇_𝑁,ℎ(𝜆𝑢)_1,Ω+𝐶𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ0,Ω. (57) Plugging (20), (52), and (53) into (57) yields (51).
If (𝜆,𝑢) is an eigenpair of (24), let 𝑢_𝑁,ℎ=𝐸_𝑁,ℎ𝑢; by the same argument we can prove (51) and (52).

4. A Posteriori Error Estimates

Based on [20], we will discuss a posteriori error estimates. We further assume that Ω⊂𝑅², the partition 𝐾_ℎ is 𝛾-shape regular, and the polynomial degree of neighboring elements are comparable; that is, there exists 𝛾>0, such that for all 𝜅,𝜅^′∈𝐾_ℎ, 𝜅∩𝜅^′≠∅, 𝛾⁻¹ℎ_𝜅≤ℎ_𝜅′≤𝛾ℎ_𝜅,𝛾⁻¹𝑁_𝜅+1≤𝑁_𝜅′+1≤𝛾𝑁_𝜅+1. (58)

We refer to the ℎ𝑝-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 𝐾_ℎ is 𝛾-shape regular and the polynomial distribution 𝐍 is comparable. Then there exists a positive constant 𝐶=𝐶(𝛾) and the clément operator 𝐼∶𝐻¹₀(Ω)→𝑆_𝑁,ℎ(Ω), such that ‖𝑣−𝐼𝑣‖_0,𝜅ℎ≤𝐶^𝜅𝑁_𝜅‖∇𝑣‖_0,𝜔𝜅,‖𝑣−𝐼𝑣‖_0,𝑒≤𝐶ℎ_𝑒𝑁_𝑒‖∇𝑣‖_0,𝜔𝑒, (59)(60) where ℎ_𝑒 is the length of the edge 𝑒 and 𝑁_𝑒=max(𝑁_𝜅1,𝑁_𝜅2), where 𝜅₁,𝜅₂ are elements sharing the edge 𝑒 and 𝜔_𝜅,𝜔_𝑒 are patches covering 𝜅 and 𝑒 with a few layers, respectively.
Define interval 𝐼̂=(0,1) and weight function Φ̂𝐼(𝑥)∶=𝑥(1−𝑥). Denote the reference square and triangle element by 𝜅=(0,1)² and 𝜅={(𝑥,𝑦)|0<𝑥<1,0<𝑦<√3(1/2−|1/2−𝑥|)}, respectively. Define weight function Φ𝜅(𝑥)∶=dist(𝑥,𝜕𝜅).

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.

Lemma 9. Let −1<𝛼<𝛽,𝜎∈[0,1]. Then there exists 𝐶=𝐶(𝛼,𝛽), such that for all 𝑁∈ℕ and all univariate polynomials 𝜋_𝑁 of degree 𝑁, 𝐼̂Φ^𝛼̂𝐼(𝑥)𝜋_𝑁(𝑥)²𝑑𝑥≤𝐶𝑁^{2(𝛽−𝛼)}𝐼̂Φ^𝛽𝐼̂𝜋_𝑁(𝑥)²𝑑𝑥. (61)

Lemma 10. Let −1<𝛼<𝛽,𝜎∈[0,1]. Then there exist 𝐶₁=𝐶(𝛼,𝛽),𝐶₂=𝐶_𝜎>0, such that for all 𝑁∈ℕ and all polynomials 𝜋_𝑁 of degree bi-𝑁, 𝜅Φ𝜅^𝛼𝜋_𝑁²𝑑𝑥𝑑𝑦≤𝐶₁𝑁^{2(𝛽−𝛼)}𝜅Φ^𝛽𝜅𝜋_𝑁²𝜅Φ𝑑𝑥𝑑𝑦,_2𝜎𝜅∇𝜋_𝑁²𝑑𝑥𝑑𝑦≤𝐶₂𝑁^2(2−𝜎)𝜅Φ𝜅^𝜎𝜋_𝑁²𝑑𝑥𝑑𝑦. (62)(63)

Lemma 11. Let 𝛼∈(1/2,1].𝑒̂∶=(0,1)×{0}, Φ𝑒̂∶=𝑥(1−𝑥); then there exists 𝐶_𝛼>0 such that for all 𝑁∈ℕ,𝜀∈(0,1], and all univariate polynomials 𝜋 of degree 𝑁, there exists an extension 𝑣𝑒̂∈𝐻¹(𝜅) and holds 𝑣𝑒|̂𝑒̂=𝜋⋅Φ𝑒^𝛼̂,𝑣𝑒|̂_𝜕𝜅_⧵𝑒̂𝑣𝑒̂=0,_20,𝜅≤𝐶_𝛼𝜀𝜋Φ𝑒̂^𝛼/22_0,𝑒,̂∇𝑣𝑒̂²_0,𝜅≤𝐶_𝛼𝜀𝑁^2(2−𝛼)+𝜀⁻¹𝜋Φ𝑒̂^𝛼/22_0,𝑒.̂ (64)(65)(66)

It is easy to know that the three lemmas above hold for complex-valued polynomials.

Let 𝐷_𝜅,𝐛_𝜅, and 𝑐_𝜅 be the interpolations of 𝐷, 𝐛, and 𝑐 in 𝜅 with the polynomial degree 𝑁_𝜅 (resp. degree 𝑁_𝜅 in every direction), respectively, or the 𝐿²(𝜅)-projection on the space of polynomials with degree 𝑁_𝜅. For convenient argument, here and hereafter we assume that (𝜆,𝑢) and (𝜆^∗=𝜆,𝑢^∗) are the eigenpairs of the eigenvalue problem (24) and its adjoint problem (31), respectively. (𝜆_𝑁,ℎ,𝑢_𝑁,ℎ) and (𝜆^∗_𝑁,ℎ=𝜆_𝑁,ℎ,𝑢^∗_𝑁,ℎ) are the solutions of the corresponding spectral element approximations (26) and (32), respectively.

Denote 𝐿_𝜅𝑢_𝑁,ℎ∶=−∇⋅𝐷_𝜅∇𝑢_𝑁,ℎ+𝐛_𝜅⋅∇𝑢_𝑁,ℎ+𝑐_𝜅𝑢_𝑁,ℎ,𝐿^∗_𝜅𝑢^∗_𝑁,ℎ∶=−∇⋅𝐷_𝜅∇𝑢^∗_𝑁,ℎ−𝐛_𝜅⋅∇𝑢^∗_𝑁,ℎ+𝑐_𝜅−∇⋅𝐛_𝜅𝑢^∗_𝑁,ℎ. (67) Define the local error indicators 𝜂²_𝛼;𝜅∶=𝜂²_{𝛼;𝐵𝜅}+𝜂²_{𝛼;𝐸𝜅},𝜂^∗2_𝛼;𝜅∶=𝜂^∗2_{𝛼;𝐵𝜅}+𝜂^∗2_{𝛼;𝐸𝜅}. (68) Their first terms 𝜂²_{𝛼;𝐵𝜅},𝜂^∗2_{𝛼;𝐵𝜅} are the weighted element internal residuals given by 𝜂²_{𝛼;𝐵𝜅}ℎ∶=^2𝜅𝑁²_𝜅−𝐿_𝜅𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎΦ_𝜅^𝛼/22_0,𝜅,𝜂^∗2_{𝛼;𝐵𝜅}ℎ∶=^2𝜅𝑁²_𝜅−𝐿^∗_𝜅𝑢^∗_𝑁,ℎ+𝜆^∗_𝑁,ℎ𝑢^∗_𝑁,ℎΦ_𝜅^𝛼/22_0,𝜅. (69) Their second terms 𝜂²_{𝛼;𝐸𝜅},𝜂^∗2_{𝛼;𝐸𝜅} are the weighted element boundary residuals given by 𝜂²_{𝛼;𝐸𝜅}∶=_{𝑒⊂𝜕𝜅∩Ω}ℎ_𝑒2𝑁_𝑒𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒,𝜂^∗2_{𝛼;𝐸𝜅}∶=_{𝑒⊂𝜕𝜅∩Ω}ℎ_𝑒2𝑁_𝑒𝐷_𝜅𝜕𝑢^∗_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒, (70) where we denote the jump of the normal derivatives of 𝑢_𝑁,ℎ and 𝑢^∗_𝑁,ℎ across the edges by [𝜕𝑢_𝑁,ℎ/𝜕𝑛] and [𝜕𝑢^∗_𝑁,ℎ/𝜕𝑛], respectively. ℎ_𝑒 is the length of edge 𝑒. The weight functions Φ_𝜅 and Φ_𝑒 are scaled transformations of the weight functions Φ𝜅 and Φ𝑒̂; that is, if 𝐹_𝜅 is the element map for element 𝜅 and 𝑒 is the image of the edge 𝑒̂ under 𝐹_𝜅, then Φ_𝜅=𝐶_𝜅Φ𝜅∘𝐹_𝜅⁻¹,Φ_𝑒=𝐶_𝑒Φ𝑒̂∘𝐹_𝜅⁻¹, (71) where we choose 𝐶_𝜅,𝐶_𝑒>0, such that _𝜅Φ_𝜅𝑑𝑥𝑑𝑦=_𝜅𝑑𝑥𝑑𝑦,_𝑒Φ_𝑒𝑑𝑠=_𝑒𝑑𝑠. (72) We define the global error indicators as follows: 𝜂²_𝛼∶=_{𝜅∈𝐾ℎ}𝜂²_𝛼;𝜅,𝜂_𝛼^∗2∶=_{𝜅∈𝐾ℎ}𝜂^∗2_𝛼;𝜅. (73)

Theorem 12. Let 𝛼∈[0,1]. Then there exists a constant 𝐶>0 independent of ℎ,𝐍, and 𝜅, such that 𝑢−𝑢_𝑁,ℎ²_1,Ω≤𝐶_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂²_𝛼;𝜅+𝐶_{𝜅∈𝐾ℎ}ℎ²_𝜅𝑁²_𝜅𝐿_𝜅𝑢_𝑁,ℎ−𝐿𝑢_𝑁,ℎ²_0,𝜅+_{𝑒⊂𝜕𝜅∩Ω}ℎ_𝑒𝑁_𝑒𝐷−𝐷_𝜅²_0,𝑒×𝜕𝑢_𝑁,ℎ𝜕𝑛²_0,∞,𝑒+𝐶𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ²_0,Ω. (74)

Proof. We denote 𝑤∶=𝑢−𝑢_𝑁,ℎ−𝐼(𝑢−𝑢_𝑁,ℎ), where 𝐼 is ℎ𝑝-clément operator given by Lemma 8. We derive from 𝐻¹₀(Ω)-elliptic of 𝑎(⋅,⋅) that 𝐶𝑢−𝑢_𝑁,ℎ²_1,Ω≤𝑎𝑢−𝑢_𝑁,ℎ,𝑤+𝑎𝑢−𝑢_𝑁,ℎ,𝐼𝑢−𝑢_𝑁,ℎ=𝜆_Ω𝑢𝑤−𝑎𝑢_𝑁,ℎ+,𝑤_Ω𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ×𝐼𝑢−𝑢_𝑁,ℎ=_Ω𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑤+_Ω𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑢−𝑢_𝑁,ℎ−𝑎𝑢_𝑁,ℎ,𝑎𝑢,𝑤_𝑁,ℎ,𝑤=_{𝜅∈𝜅ℎ𝜅}𝐿𝑢_𝑁,ℎ𝑤−_{𝜅∈𝜅ℎ𝜕𝜅}𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤=_{𝜅∈𝜅ℎ𝜅}𝐿𝑢_𝑁,ℎ𝑤−12_{𝜅∈𝜅ℎ𝑒⊂𝜕𝜅∩Ω𝑒}𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤. (75) Therefore, 𝐶𝑢−𝑢_𝑁,ℎ²_1,Ω≤_{𝜅∈𝐾ℎ𝜅}−𝐿𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑤+12_{𝜅∈𝐾ℎ𝑒⊂𝜕𝜅∩Ω𝑒}𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤+_Ω𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑢−𝑢_𝑁,ℎ, (76) which together with _𝑒𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤=_𝑒𝐷−𝐷_𝜅𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤+_𝑒𝐷_𝜅𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤,_𝜅−𝐿𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑤=_𝜅−𝐿_𝜅𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑤+_𝜅𝐿_𝜅𝑢_𝑁,ℎ−𝐿𝑢_𝑁,ℎ𝑤 (77) and using Cauchy-Schwartz inequality, the ℎ𝑝-clément interpolation estimates in Lemma 8 then yield 𝑢−𝑢_𝑁,ℎ²_1,Ω≤𝐶_{𝜅∈𝐾ℎ}𝜂²_0;𝐵𝜅+𝜂²_0;𝐸𝜅+ℎ²_𝜅𝑁²_𝜅𝐿_𝜅𝑢_𝑁,ℎ−𝐿𝑢_𝑁,ℎ²_0,𝜅+_{𝑒⊂𝜕𝜅∩Ω}ℎ_𝑒𝑁_𝑒𝐷−𝐷_𝜅𝜕𝑢_𝑁,ℎ𝜕𝑛²_0,𝑒^1/2×𝑢−𝑢_𝑁,ℎ1,Ω+𝐶𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ0,Ω𝑢−𝑢_𝑁,ℎ0,Ω. (78)
Using scaled transformation and setting 𝛼=0,𝛽=𝛼 in (61) and (62), we get 𝜂_0;𝐸𝜅≤𝐶𝑁^𝛼_𝜅𝜂_{𝛼;𝐸𝜅} and 𝜂_0;𝐵𝜅≤𝐶𝑁^𝛼_𝜅𝜂_{𝛼;𝐵𝜅}; then this proof concludes.

For the adjoint eigenvalue problem, we still have the following.

Theorem 13. Let 𝛼∈[0,1]. Then there exists a constant 𝐶>0 independent of ℎ,𝐍, and 𝜅, such that 𝑢^∗−𝑢^∗_𝑁,ℎ²_1,Ω≤𝐶_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂^∗2_𝛼;𝜅+𝐶_{𝜅∈𝐾ℎ}ℎ²_𝜅𝑁²_𝜅𝐿^∗_𝜅𝑢^∗_𝑁,ℎ−𝐿^∗𝑢^∗_𝑁,ℎ²+_{𝑒⊂𝜕𝜅∩Ω}ℎ_𝑒𝑁_𝑒𝐷−𝐷_𝜅²_0,𝑒𝜕𝑢^∗_𝑁,ℎ𝜕𝑛²_0,∞,𝑒+𝐶𝜆^∗𝑢^∗−𝜆^∗_𝑁,ℎ𝑢^∗_𝑁,ℎ²_0,Ω. (79)

Lemma 14. Let 𝛼∈[0,1],𝜀>0. Then there exists a constant 𝐶(𝜀)>0 independent of ℎ,𝐍, and 𝜅, such that 𝜂²_{𝛼;𝐵𝜅}≤𝐶(𝜀)𝑁_𝜅^2(1−𝛼)𝑢−𝑢_𝑁,ℎ²_1,𝜅+𝑁_𝜅^{max{1+2𝜀−2𝛼,0}}ℎ²_𝜅𝑁²_𝜅×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²_0,𝑘. (80)

Proof. We denote 𝑣_𝜅∶=(−𝐿_𝜅𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ)Φ^𝛼_𝜅∈𝐻¹₀(𝜅) with 𝛼∈(0,1] and extend 𝑣_𝜅 to Ω by 𝑣_𝜅=0 on Ω⧵𝜅; then 𝑣_𝜅Φ_𝜅^−𝛼/22_0,𝜅=_𝜅−𝐿_𝜅𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ𝑣_𝜅=−_𝜅𝐿_𝜅𝑢_𝑁,ℎ𝑣_𝜅+𝑎𝑢,𝑣_𝜅+_𝜅𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢𝑣_𝜅=𝑎𝑢−𝑢_𝑁,ℎ,𝑣_𝜅+_𝜅𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ𝑣_𝜅≤𝐶𝑢−𝑢_{𝑁,ℎ1,𝜅}𝑣_𝜅1,𝜅+𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎΦ_𝜅^𝛼/2_0,𝜅×𝑣_𝜅Φ_𝜅^−𝛼/2_0,𝜅. (81) We consider the 𝐻¹ semi norm for 𝑣_𝜅. Using the polynomial inverse estimates (62)-(63) in Lemma 10, by transformation between the reference element 𝜅 and 𝜅, we find for 𝛼>1/2 that 𝑣_𝜅²_1,𝜅≤2_𝜅Φ_𝜅^2𝛼∇𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²+2_𝜅∇Φ^𝛼_𝜅²𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²𝑁≤𝐶^{𝜅2(2−𝛼)}ℎ²_𝜅^𝜅Φ^𝛼_𝜅𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²1+𝐶ℎ²_𝜅^𝜅Φ_𝜅^2(𝛼−1)𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²𝑁≤𝐶^{𝜅2(2−𝛼)}ℎ²_𝜅^𝜅Φ^𝛼_𝜅𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²=𝐶𝑁_𝜅^2(1−𝛼)𝑁²_𝜅ℎ²_𝜅‖𝑣_𝜅Φ_𝜅^−𝛼/2‖²_0,𝜅. (82) Note that (62) is applicable since 𝛼>1/2 implies 2(𝛼−1)>−1; thus, we set 𝛽=𝛼,𝛼=2(𝛼−1) in (62); then the third inequality above holds.
Since 𝜂_{𝛼;𝐵𝜅}=ℎ_𝜅/𝑁_𝜅‖𝑣_𝜅Φ_𝜅^−𝛼/2‖_0,𝜅, we obtain 𝜂_{𝛼;𝐵𝜅}≤𝐶𝑁_𝜅^1−𝛼𝑢−𝑢_{𝑁,ℎ1,𝜅}+ℎ_𝜅𝑁_𝜅𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ+𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢_0,𝜅. (83) To obtain an upper bound in the case of 0≤𝛼≤1/2, we use the polynomial inverse estimate (62) in Lemma 10; for 𝛽>1/2, we derive from (62) that 𝑁_𝜅ℎ_𝜅𝜂_{𝛼;𝐵𝜅}=Φ_𝜅^𝛼/2𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_{𝑁,ℎ0,𝜅}≤𝐶𝑁_𝜅^𝛽−𝛼𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎΦ_𝜅^𝛽/2_0,𝜅=𝐶𝑁_𝜅^𝛽−𝛼𝑁_𝜅ℎ_𝜅𝜂_{𝛽;𝐵𝜅}≤𝐶𝑁_𝜅^𝛽−𝛼𝑁_𝜅^1−𝛽𝑁_𝜅ℎ_𝜅𝑢−𝑢_{𝑁,ℎ1,𝜅}+𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_{𝑁,ℎ0,𝜅}. (84) Setting 𝛽=1/2+𝜀,𝜀>0, 𝜂_{𝛼;𝐵𝜅}≤𝐶(𝜀)𝑁_𝜅^1−𝛼𝑢−𝑢_{𝑁,ℎ1,𝜅}+𝑁_𝜅^{1/2+𝜀−𝛼}ℎ_𝜅𝑁_𝜅×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_{𝑁,ℎ0,𝜅}. (85) We obtain the desired result immediately from (83) and (85).

In order to obtain a local upper bound for the error indicator 𝜂_𝛼;𝜅, we consider the edge residual term 𝜂_{𝛼;𝐸𝜅}. we introduce the set 𝜔_𝜅∶=∪𝜅^′𝜅^′and𝜅shareatleastoneedge. (86)

Lemma 15. Let 𝛼∈[0,1],𝜀>0. Then there exists a constant 𝐶(𝜀)>0 independent of ℎ,𝐍, and 𝜅, such that 𝜂²_{𝛼;𝐸𝜅}≤𝐶(𝜀)𝑁_𝜅^{max(1−2𝛼+2𝜀,0)}×𝑁_𝜅𝑢−𝑢_𝑁,ℎ²_1,𝜔𝜅+𝑁_𝜅^2𝜀ℎ²_𝜅𝑁²_𝜅×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²_0,𝜔𝜅. (87)

Proof. We will use weight functions on edge and a suitable extension operator. For a given element 𝜅 with edge 𝑒, we choose the element 𝜅₁ so that 𝜕𝜅₁∩𝜕𝜅=𝑒. Denote 𝜅_𝑒∶=𝜅₁∪𝜅; we construct a function 𝑤_𝑒∈𝐻¹₀(𝜅_𝑒) with 𝑤_𝑒|_𝑒=𝐷_𝜅[𝜕𝑢_𝑁,ℎ/𝜕𝑛]Φ^𝛼_𝑒 as follows.
Let 𝑣𝑒̂=𝐶_𝑒𝐷_𝜅[𝜕𝑢_𝑁,ℎ/𝜕𝑛]Φ𝑒^𝛼̂ (𝐶_𝑒 is defined by (71)). Using Lemma 11, we extend 𝑣𝑒̂ to 𝜅, where the polynomial 𝜋 corresponds to 𝐶_𝑒𝐷_𝜅[𝜕𝑢_𝑁,ℎ/𝜕𝑛]. Define 𝑤_𝑒|_𝜅 and 𝑤_𝑒|_𝜅1 as the affine transformation of 𝑣𝑒̂ in 𝜅; Thus, 𝑤_𝑒 is a piecewise 𝐻¹-function. From (64), we know 𝑤_𝑒 vanishes on 𝜕𝜅_𝑒; Therefore, 𝑤_𝑒∈𝐻¹₀(𝜅_𝑒). It is trivial to extend 𝑤_𝑒 to Ω, such that 𝑤_𝑒=0 in Ω⧵𝜅_𝑒. We find 𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒=_𝑒𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤_𝑒+_𝑒𝐷_𝜅−𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤_𝑒=_𝜅𝑒𝐿𝑢_𝑁,ℎ𝑤_𝑒−𝑎𝑢_𝑁,ℎ,𝑤_𝑒+_𝑒𝐷_𝜅−𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤_𝑒=_𝜅𝑒𝐿𝑢_𝑁,ℎ−𝜆𝑢𝑤_𝑒+𝑎𝑢−𝑢_𝑁,ℎ,𝑤_𝑒+_𝑒𝐷_𝜅−𝐷𝜕𝑢_𝑁,ℎ𝜕𝑛𝑤_𝑒≤𝐿𝑢_𝑁,ℎ−𝜆𝑢_0,𝜅𝑒𝑤_{𝑒0,𝜅𝑒}+𝐶𝑢−𝑢_{𝑁,ℎ1,𝜅𝑒}𝑤_{𝑒1,𝜅𝑒}+𝐷_𝜅−𝐷𝐷_{𝜅0,∞,𝑒}𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒. (88) Therefore, 𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒≤𝐶𝐿𝑢_𝑁,ℎ−𝜆𝑢_0,𝜅𝑒𝑤_{𝑒0,𝜅𝑒}+𝐶𝑢−𝑢_{𝑁,ℎ1,𝜅𝑒}𝑤_{𝑒1,𝜅𝑒}. (89) We consider the case of 𝛼∈(1/2,1] first. Using the affine equivalence and (65)-(66) in Lemma 11, we obtain the upper bounds for ‖𝑤_𝑒‖_0,𝜅𝑒 and |𝑤_𝑒|_1,𝜅𝑒 as follows: 𝑤_𝑒²_1,𝜅𝑒1≤𝐶ℎ_𝜅𝜀𝑁_𝜅^2(2−𝛼)+𝜀⁻¹×𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒,𝑤_{𝑒0,𝜅𝑒}≤𝐶ℎ_𝜅𝜀𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/22_0,𝑒. (90) It follows from (89)-(90) that 𝐷_𝜅𝜕𝑢_𝑁,ℎΦ𝜕𝑛^𝑒𝛼/2_0,𝑒1≤𝐶ℎ_𝜅𝜀𝑁_𝜅^2(2−𝛼)+𝜀^−11/2×𝑢−𝑢_{𝑁,ℎ1,𝜅𝑒}+ℎ_𝜅𝜀^1/2‖𝐿𝑢_𝑁,ℎ−𝜆𝑢_0,𝜅𝑒. (91) By the definition of 𝜂²_{𝛼;𝐸𝜅} and setting 𝛼=0 in Lemma 14, we get 𝐿_𝜅𝑢_𝑁,ℎ−𝜆_𝑁,ℎ𝑢_𝑁,ℎ²_0,𝑘(𝜀𝑁≤𝐶)^4𝜅ℎ²_𝜅𝑢−𝑢_𝑁,ℎ²_1,𝑘+𝑁_𝜅^1+2𝜀×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²_0,𝑘, (92) by the triangle inequality 𝐿𝑢_𝑁,ℎ−𝜆𝑢_0,𝜅𝑒≤𝐿_𝜅𝑢_𝑁,ℎ−𝜆_𝑁,ℎ𝑢_{𝑁,ℎ0,𝜅𝑒}+𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_{𝑁,ℎ0,𝜅𝑒}. (93) Combining the three inequalities above and summing, we have 𝜂²_{𝛼;𝐸𝜅}1≤𝐶𝑁_𝜅𝜀𝑁_𝜅^2(2−𝛼)+𝜀⁻¹+𝑁³_𝜅𝜀×𝑢−𝑢_𝑁,ℎ²_1,𝜔𝜅+𝐶𝜀𝑁_𝜅^2(1+𝜀)ℎ²_𝜅𝑁²_𝜅×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²_0,𝜔𝜅. (94)
Setting 𝜀=1/𝑁²_𝜅 in the above inequality yields the assertion for 𝛼>1/2. For the case of 𝛼∈[0,1/2], we set 𝛽=1/2+𝜀, use (62) in Lemma 10 to get 𝜂_{𝛼;𝐸𝜅}≤𝑁_𝜅^𝛽−𝛼𝜂_{𝛽;𝐸𝜅}, and find the desired result.

Combining Lemmas 14 and 15, we obtain the following theorem.

Theorem 16. Let 𝛼∈[0,1],𝜀>0. Then there exists a constant 𝐶>0 independent of ℎ,𝐍, and 𝜅, such that 𝜂²_𝛼;𝜅≤𝐶(𝜀)𝑁_𝜅^{max(1−2𝛼+2𝜀,0)}×𝑁_𝜅𝑢−𝑢_𝑁,ℎ²_1,𝜔𝜅+𝑁_𝜅^2𝜀ℎ²_𝜅𝑁²_𝜅×𝜆_𝑁,ℎ𝑢_𝑁,ℎ−𝜆𝑢+𝐿𝑢_𝑁,ℎ−𝐿_𝜅𝑢_𝑁,ℎ²_0,𝜔𝜅. (95)

Similarly, we have Theorem 17.

Theorem 17. Let 𝛼∈[0,1],𝜀>0. Then there exists a constant 𝐶>0 independent of ℎ,𝐍, and 𝜅, such that 𝜂^∗2_𝛼;𝜅≤𝐶(𝜀)𝑁_𝜅^{max(1−2𝛼+2𝜀,0)}×𝑁_𝜅𝑢^∗−𝑢^∗_𝑁,ℎ²_1,𝜔𝜅+𝑁_𝜅^2𝜀ℎ²_𝜅𝑁²_𝜅×𝜆^∗_𝑁,ℎ𝑢^∗_𝑁,ℎ−𝜆^∗𝑢^∗+𝐿^∗𝑢^∗_𝑁,ℎ−𝐿^∗_𝜅𝑢^∗_𝑁,ℎ²_0,𝜔𝜅. (96)

In order to estimate bounds of |𝜆−𝜆_𝑁,ℎ|, we also need Lemma 18 (see [8, 10]).

Lemma 18. Let (𝜆,𝑢) be an eigenpair of (24), and let (𝜆^∗=𝜆,𝑢^∗) be the associated eigenpair of the adjoint problem (31). Then for all 𝑤,𝑤^∗∈𝐻¹₀(Ω), (𝑤,𝑤^∗)≠0, 𝑎𝑤,𝑤^∗(𝑤,𝑤^∗)=𝑎−𝜆𝑤−𝑢,𝑤^∗−𝑢^∗(𝑤,𝑤^∗)−𝜆𝑤−𝑢,𝑤^∗−𝑢^∗(𝑤,𝑤^∗). (97)

Theorem 19. Under the assumptions of Theorem 7, we assume that 𝐷,𝐛, and 𝑐 are smooth enough, and let 𝛼∈[0,1]. Then there exists an eigenpair (𝜆,𝑢) of (24), such that 𝑢−𝑢_𝑁,ℎ1,Ω≤𝐶_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂²_𝛼;𝜅^1/2,𝜂²_𝛼;𝜅≤𝐶(𝜀)𝑁_𝜅^{max(2−2𝛼+2𝜀,1)}𝑢−𝑢_𝑁,ℎ²_1,𝜔𝜅. (98)(99) Further let the ascent of 𝜆_𝑁,ℎ be 𝑙=1, and let (𝜆^∗_𝑁,ℎ,𝑢^∗_𝑁,ℎ) be the corresponding adjoint eigenpair of (32), then there exists an adjoint eigenpair (𝜆^∗,𝑢^∗) of (31), such that 𝜆_𝑁,ℎ−𝜆≤𝐶_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂²_𝛼;𝜅^1/2_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂^∗2_𝛼;𝜅^1/2. (100) Particularly, if the eigenvalue problem (23) is symmetric (i.e., 𝐛=0), then 𝐶𝐶(𝜀)⁻¹_{𝜅∈𝐾ℎ}𝑁_𝜅^{min(2𝛼−2−2𝜀,−1)}𝜂²_𝛼;𝜅≤𝜆_𝑁,ℎ−𝜆. (101)

Proof. We know from the assumption 𝐷,𝑐∈𝐻^𝑡1(𝜅),𝐛∈(𝐻^𝑡1(𝜅))². By the interpolation error estimates (14) and (15), we have 𝐿_𝜅𝑢_𝑁,ℎ−𝐿𝑢_{𝑁,ℎ0,𝜅}≤𝐶ℎ^{min(𝑁𝜅+1,𝑡1}_𝜅⁾⁻¹𝑁^−𝑡1_𝜅⁺¹. (102) From 𝐷∈𝐻^𝑡1(𝜅), we know that 𝐷∈𝐻^𝑡1−1/2(𝑒). By the interpolation error estimate on edge of element (see formula (5.4.42) in [2]), we get 𝐷−𝐷_𝜅0,𝑒≤𝐶ℎ^{min(𝑁𝜅+1,𝑡1}_𝑒^−1/2)𝑁^−𝑡1_𝜅^+1/2. (103) Note that the formula (51) gives the optimal orders of convergence; thus, we deduce that the second and third terms on the right side of (74) are higher order infinitesimals. We derive from (52) and (53), and 𝑁=𝑁_𝜅, that 𝜆𝑢−𝜆_𝑁,ℎ𝑢_𝑁,ℎ0,Ω≤𝜆−𝜆_𝑁,ℎ‖𝑢‖_0,Ω+𝜆_𝑁,ℎ𝑢−𝑢_𝑁,ℎ0,Ω≤𝐶ℎ^{𝜏1+𝜏2−2}𝑁^{𝑡1+𝑡2−2}+𝐶ℎ^{𝑟2+𝜏1−1}𝑁^{𝑟2+𝑡1−1}≤𝐶ℎ^{𝑟2+𝜏1−1}𝑁^{𝑟2+𝑡1−1}. (104) Therefore, the fourth term on the right side of (74) is also a higher order infinitesimal. Up to higher order terms, we get (98). We ignore higher order infinitesimals in (95) and get (99). From Lemma 4 in [10], we know that (𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ)=1 and 𝑢^∗_𝑁,ℎ is uniformly bounded with ℎ and 𝑁. By the same argument of (98), we can deduce that 𝑢^∗−𝑢^∗_𝑁,ℎ²_1,Ω≤𝐶_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂^∗2_𝛼;𝜅. (105) From (97), we have 𝑎𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ=𝑎𝑢−𝜆_𝑁,ℎ−𝑢,𝑢^∗_𝑁,ℎ−𝑢^∗𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ𝑢−𝜆^𝑁,ℎ−𝑢,𝑢^∗_𝑁,ℎ−𝑢^∗𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ; (106) that is, 𝜆_𝑁,ℎ−𝜆=𝑎𝑢_𝑁,ℎ−𝑢,𝑢^∗_𝑁,ℎ−𝑢^∗−𝜆𝑢_𝑁,ℎ−𝑢,𝑢^∗_𝑁,ℎ−𝑢^∗. (107) Substituting (98) and (105) into the above equality, we obtain (100).
If the eigenvalue problem (23) is symmetric (i.e., 𝐛=0), then 𝜆_𝑁,ℎ−𝜆=𝑎𝑢_𝑁,ℎ−𝑢,𝑢_𝑁,ℎ𝑢−𝑢−𝜆_𝑁,ℎ−𝑢,𝑢_𝑁,ℎ−𝑢. (108) Up to higher order term 𝜆(𝑢_𝑁,ℎ−𝑢,𝑢_𝑁,ℎ−𝑢), by (99) we get (101).

Remark 20. Babuška and Osborn [17] have discussed hp finite element approximation with simplex partition for eigenvalue problems. Obviously, the Interpolation estimates (14) and (15) hold for hp finite element with simplex partition (see [24]). Therefore, our theoretical results of spectral methods and spectral methods for eigenvalue problems, which have been discussed in Sections 3 and 4, hold for hp finite element with simplex partition.

5. 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.

In our experiment, we compute 1/|(𝑢_𝑁,ℎ,𝑢^∗_𝑁,ℎ)| as condition number for simple eigenvalue (see Remark 2.1 in [25]), where 𝑢_𝑁,ℎ and 𝑢^∗_𝑁,ℎ are eigenfunctions of eigenvalue problem (25) and its adjoint problem (32) normalized with ‖⋅‖_0,Ω, respectively.

5.1. Example 1

Consider the nonsymmetric eigenvalue problem −Δ𝑢+10𝑢_𝑥+𝑢_𝑦=𝜆𝑢,inΩ=(0,1)²,𝑢=0,on𝜕Ω. (109)

The first eigenvalue of (109) 𝜆₁=101/4+2𝜋² is a simple eigenvalue. And the corresponding eigenfunctions are sufficiently smooth.

5.1.1. 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 𝑁>19 for Eq-SM. Table 1 tells us that when 𝑁>11, the accuracy of first eigenvalue obtained by Eq-SM is not as good as obtained that by LGL-SM and Modal-SM. When 𝑁=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.

Table 1 

Errors of LGL-SM, modal, and Eq-SM for 1st eigenvalue.

Figure 1 

Condition number of first eigenvalue for SM.

5.1.2. LGL-SM and Modal-SM versus hp-SEM

Tables 1 and 2 indicate that increasing the polynomial degree 𝑁 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 2.8𝐸−14 and 1.3𝐸−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.

Table 2 

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

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.

5.1.3. 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 𝑁 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.

Table 3 

Errors of hp-FEM for the first eigenvalue.

Table 4 

Condition number of first eigenvalue for hp-SEM.

Remark 21. Condition numbers of 1st eigenvalue for hp-FEM (not listed in Table 4) are almost the same to those for hp-SEM.

5.1.4. Validity of the Error Indicator

Denote 𝜓_𝛼=_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂²_𝛼;𝜅^1/2_{𝜅∈𝐾ℎ}𝑁_𝜅^2𝛼𝜂^∗2_𝛼;𝜅^1/2. (110)

From Theorem 19, we know that 𝜓_𝛼 is a reliable error indicator for 𝜆_𝑁,ℎ. We choose 𝜓₀ (setting 𝛼=0 in (110)) as a posteriorii error indicator.

In Figures 2 and 3, we denote the true error and est. error with |𝜆_𝑁,ℎ−𝜆| and 𝜓₀, respectively.

Figure 2 

The Error indicator 𝝍𝟎 of LGL-SM.

Figure 3 

The Error indicator 𝝍𝟎 of hp-SEM (ℎ=1/2).

As is depicted in Figure 2, when the polynomial degree 𝑁≤12, the error indicator 𝜓₀ 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 𝜓₀ shows almost the same algebraic decay as the true error with the polynomial degree 𝑁 (≤12) increasing. Nevertheless, the error indicator 𝜓₀ cannot approximate the true errors if 𝑁 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 𝜓₀ and the true errors for hp-SEM.

5.2. Example 2

Consider the nonsymmetric eigenvalue problem −Δ𝑢+10𝑢_𝑥()=𝜆𝑢,inΩ=−1,1²(0,1)²,𝑢=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 𝜓₀ for reference.

Table 5 

The Approximate eigenvalues and indicator 𝜓₀ of P-SEM.

5.2.1. 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 𝑁 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.

Figure 4 

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

Figure 5 

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

5.2.2. 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 𝑁=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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11161012) and the Educational Administration and Innovation Foundation of Graduate Students of Guizhou Normal University (no. 2012(11)).