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

Authors: Jiayu Han; Yidu Yang
Journal: Abstract and Applied Analysis
 (2013)

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## 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 derivedp-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.


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## Body

## 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 derivedp-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. Section2 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(1)Lw=-∇·(D∇w)+b·∇w+cw=f,inΩ,w=0,on∂Ω,
where Ω⊂Rd(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 Hm(Ω), H01(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 forma(·,·) as follows:
(2)a(w,v)=∫Ω‍D∇w∇v¯+(b·∇w)v¯a(w,v)+cwv¯,∀w,v∈H01(Ω).We assume thatf∈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
(3)-12∇·b+c≥0,inΩ.Under these assumptions, the bilinear forma(·,·) is continuous in H01(Ω) and H01(Ω)-elliptic; that is, there exist two constants A,B>0 independent of w,v such that
(4)|a(w,v)|≤A∥w∥1,Ω∥v∥1,Ω,∀w,v∈H01(Ω),Rea(v,v)≥B∥v∥1,Ω2,∀v∈H01(Ω).The corresponding variational formulation of (1) is given as follows: find w∈H01(Ω), such that
(5)a(w,v)=(f,v),∀v∈H01(Ω).The adjoint problem of (5) is as follows: find w*∈H01(Ω), such that
(6)a(v,w*)=(v,f),∀v∈H01(Ω).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
(7)∥w∥r1+1,Ω≤C∥f∥0,Ω,0<r1≤1,(8)∥w*∥r2+1,Ω≤C∥f∥0,Ω,0<r2≤1.We assume thatKh={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={Nκ}, where Nκ is the polynomial degree in κ. Let hκ be the diameter of the element κ, and let h=maxκ∈Khhκ.We define spectral and spectral element spaces as follows:(9)SN(Ω)={v∈PN(Ω),v|∂Ω=0},SN,h(Ω)={v∈C(Ω¯):v|κ∈PNκ(κ),∀κ∈Kh,v|∂Ω=0},
where PN(Ω) and PNκ(κ) are polynomial spaces of degree N (resp. degree N in every direction) in Ω and degree Nκ (resp. degree Nκ in every direction) in the element κ, respectively.The spectral approximation of (5) is as follows: find wN∈SN(Ω), such that
(10)a(wN,v)=(f,v),∀v∈SN(Ω).The spectral element approximation of (5) is as follows: find wN,h∈SN,h(Ω), such that
(11)a(wN,h,v)=(f,v),∀v∈SN,h(Ω).We assume thatf∈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(12)INκ,hκ:C(κ)⟶PNκ(κ),IN:C(Ω)⟶SN(Ω),
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(13)IN,h:C(Ω)⟶SN,h(Ω),satisfyingIN,h|κ=INκ,hκ.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 allv∈Hsκ(κ),sκ≥(d+1)/2,
(14)∥v-INκ,hκv∥1,κ≤C(sκ)hκmin(Nκ+1,sκ)-1Nκ-sκ+1∥v∥sκ,κ,(15)∥v-INκ,hκv∥0,κ≤C(sκ)hκmin(Nκ+1,sκ)Nκ-sκ∥v∥sκ,κ.For allv∈Hs(Ω),s≥(d+1)/2,
(16)∥v-INv∥1,Ω≤C(s)N-s+1∥v∥s,Ω,(17)∥v-INv∥0,Ω≤C(s)N-s∥v∥s,Ω.We assume that the solution of boundary value problem (5) w∈H01(Ω)∩Hm(Ω)(m>1), that wN and wN,h are the solutions of (10) and (11), respectively; then we derive from Céa lemma and the interpolation estimates that
(18)∥wN-w∥1,Ω≤C(m)N-m+1∥w∥m,Ω,(19)∥wN,h-w∥1,Ω≤{∑κC(sκ)hκ2{min(Nκ+1,sκ)-1}i×Nκ2(-sκ+1)∥w∥sκ,κ2∑κC(sκ)hκ2{min(Nκ+1,sκ)-1}}1/2,
where sκ≥(d+1)/2,∀κ∈Kh.Particularly, ifNκ=N,∀κ∈Kh, then we have
(20)∥wN,h-w∥1,Ω≤C(m)hmin(N+1,m)-1×N1-m∥w∥m,Ω.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,
(21)∥wN-w∥0,Ω≤CN-m-r2+1∥w∥m,Ω,(22)∥wN,h-w∥0,Ω≤CN-m-r2+1hr2+min(N+1,m)-1∥w∥m,Ω.


## 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):
(23)Lu=λu,inΩ,u=0,on∂Ω.The variational formation of (23) is given by the following: find λ∈ℂ,0≠u∈H01(Ω), such that
(24)a(u,v)=λ(u,v),∀v∈H01(Ω).The spectral approximation scheme of (24) is given by the following: find λN∈ℂ,0≠uN∈SN(Ω), such that
(25)a(uN,vN)=λN(uN,vN),∀vN∈SN(Ω).The spectral element approximation scheme of (24) is given by the following: find λN,h∈ℂ,0≠uN,h∈SN,h(Ω), such that
(26)a(uN,h,vN,h)=λN,h(uN,h,vN,h),∀vN,h∈SN,h(Ω),Define the solution operatorsT:L2(Ω)→H01(Ω),TN:L2(Ω)→SN(Ω), and TN,h:L2(Ω)→SN,h(Ω) by
(27)a(Tf,v)=(f,v),∀f∈L2(Ω),∀v∈H01(Ω),cccicia(TNf,vN)=(f,vN),∀f∈L2(Ω),ffffffffffffffffffffffffffgffffff∀vN∈SN(Ω),22iiia(TN,hf,vN,h)=(f,vN,h),∀f∈L2(Ω),ddddddddddgddffffdd∀vN,h∈SN,h(Ω).
Obviously (see [17]), the equivalent operator forms for (24) and (26) are the following.Findλ∈ℂ,0≠u∈H01(Ω), such that
(28)Tu=λ-1u.FindλN,h∈ℂ,0≠uN,h∈SN,h(Ω), such that
(29)TN,huN,h=λN,h-1uN,h.
The adjoint problem of the eigenvalue problem (23) is
(30)L*u*=λ*u*,inΩ,u*=0,on∂Ω,
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*∈H01(Ω), such that
(31)a*(u*,v):=a(v,u*)¯=λ*(u*,v),∀v∈H01(Ω).The spectral element approximation scheme of (31) is given by the following: find λN,h*∈ℂ,0≠uN,h*∈SN,h(Ω), such that
(32)a*(uN,h*,vN,h)=λN,h*(uN,h*,vN,h),∀vN,h∈SN,h(Ω).We can likewise define the equivalent operator forms for the eigenvalue problems (31) and (32) as
(33)T*u*=λ*-1u*,TN,h*uN,h*=λN,h*-1uN,h*.Letλ 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 λN,h, λ* and λN,h* can be defined.Letλ be an eigenvalue of (23) with the algebraic multiplicity q and the ascent μ. Assume ∥TN,h-T∥1,Ω→0(N→∞,h→0); then there are eigenvalues λj,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 MN,h(λ) be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of TN,h that converge to λ.In view of adjoint problems (31) and (32), the definitions of M*(λ*) and MN,h*(λ*) are analogous to M(λ) and MN,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 whenb=0, both (24) and (26) are symmetric. Thus, the ascent μ=1 of λ, and the ascent l=1 of λN,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 finenessh not considered.Assume thatR and U are two closed subspace in H01(Ω).Denote(34)δ(R,U)=supv∈R∥v∥1,Ω=1dist(v,U),θ(R,U)=max(δ(R,U),δ(U,R)).
We say that θ(R,U) is the gap between R and U.Denote(35)εN,h=εN,h(λ)=supu∈M^(λ)infv∈SN,h(Ω)∥u-v∥1,Ω,εN,h*=εN,h*(λ*)=supu∈M*^(λ*)infv∈SN,h(Ω)∥u-v∥1,Ω.We give the following four lemmas from Theorem 8.1–8.4 in [17], which are applications to spectral element methods.Lemma 1.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0). For small enough h and big enough N, there holds
(36)θ(M(λ),MN,h(λ))≤CεN,h.Lemma 2.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0); then
(37)|λ-1-1q∑j=1q‍λj,N,h-1|≤CεN,hεN,h*.Lemma 3.
Assume that∥TN,h-T∥1,Ω→0(N→∞,h→0); then there holds
(38)|λ-λj,N,h|≤C(εN,hεN,h*)1/μ(j=1,2,…,q).
Since ker((λ-1-T)l)(l≥1) is a finite dimensional space, there exists a direct-sum decomposition H01(Ω)=ker((λ-1-T)l)⊕Ml. We define the operator El as a projection along Ml from H01(Ω) to ker((λ-1-T)l).Lemma 4.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0). Let λN,h be an eigenvalue of TN,h and limN→∞,h→0λN,h=λ. uN,h satisfies (λN,h-1-TN,h)kuN,h=0 and ∥uN,h∥1,Ω=1, where k≤μ is a positive integer. Then, for every integer l∈[k,μ], there holds
(39)∥uN,h-EluN,h∥1,Ω≤CεN,h(l-k+1)/μ.
We assume that in this section, for the sake of simplicity,Nκ=N,∀κ∈Kh.Theorem 5.
IfM(λ)⊂Ht1(Ω) and M*(λ*)⊂Ht2(Ω), then there holds the following error estimates:
(40)|1q∑j=1q‍λj,N,h-λ|≤C(hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω,(41)|λj,N,h-λ|≤C((hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω)1/μdddddddddddddddddddddddd(j=1,2,…,q),(42)θ(M(λ),MN,h(λ))≤Chτ1-1Nt1-1supu∈M^(λ)∥u∥t1,Ω
Let ∥uN,h∥1,Ω=1, and let (λN,h-1-TN,h)l1uN,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
(43)∥uN,h-u′∥1,Ω≤C((hτ1-1Nt1-1)supu∈M^(λ)∥u∥t1,Ω)(l2-l1+1)/μ,
where τ1=min(N+1,t1), τ2=min(N+1,t2).Proof.
We derive from the error estimate (20) that
(44)∥TN,h-T∥1,Ω=supf∈H01(Ω)∥(T-TN,h)f∥1,Ω∥f∥1,Ω≤C(1+r1)hr1N-r1⟶0(N⟶∞,h⟶0).
By (14),
(45)εN,h=εN,h(λ)=supu∈M^(λ)infv∈SN,h(Ω)∥u-v∥1,Ω≤C(hτ1-1Nt1-1)supu∈M^(λ)∥u∥t1,Ω.
Analogically,
(46)εN,h*≤C(hτ2-1Nt2-1)supu∈M*^(λ*)∥u∥t2,Ω.
Plugging the two inequalities above into (36), (38), and (39) yields (42), (41), and (43), respectively. We find from (37) that
(47)|1q∑j=1q‍λj,N,h-λ|=|1q∑j=1q‍λj,N,h-1-λ-1λ-1λj,N,h-1|≤C|1q∑j=1q‍λj,N,h-1-λ-1|≤CεN,hεN,h*,
combining with (45) and (46) yields (40).Supposing that∥TN,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(48)R(z)=(T-z)-1,R(TN,h,z)=(TN,h-z)-1.Define the spectral projection operators(49)E=-12iπ∫Γ‍R(T,z)dz:H01(Ω)⟶M(λ),EN,h=-12iπ∫Γ‍R(TN,h,z)dz:H01(Ω)⟶MN,h(λ).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∥TN,h-T∥0,Ω→0(N→∞,h→0), then there holds that EN,h→E(p), R(TN,h,z) is uniformly bounded with N and h, and
(50)∥(EN,h-E)v∥0,Ω≤Cmaxz∈Γ∥(T-TN,h)R(z)v∥0,Ω,fffffffffffffddddddddgddff∀v∈H01(Ω),∥(EN,h-E)v∥0,Ω≤Cmaxz∈Γ∥(T-TN,h)R(TN,h,z)v∥0,Ω,ddddddddddddddddffffffffgfffffffffffffdddd∀v∈H01(Ω).Theorem 7.
Under the assumptions of Theorem5, further assume that the ascent of λ is μ=1. Let (λN,h,uN,h) be an eigenpair of (26) with ∥uN,h∥0,Ω=1; then there exists an eigenpair (λ,u) of (24), such that
(51)∥uN,h-u∥1,Ω≤Chτ1-1∥u∥t1,ΩNt1-1,(52)∥uN,h-u∥0,Ω≤Chr2+τ1-1∥u∥t1,ΩNr2+t1-1,(53)|λN,h-λ|≤C((hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω),
where τ1=min(N+1,t1) and τ2=min(N+1,t2).
Let(λ,u) be an eigenpair of (24). If λN,h is an eigenvalue of (26) convergence to λ, then there exists uN,h∈ker(λN,h-1-TN,h), such that (51)–(53) hold.Proof.
We deduce (53) immediately from (41). We derive from (22) and (7) that
(54)∥Tf-TN,hf∥0,Ω≤CN-r1-r2hr1+r2∥f∥0,Ω;
thus,∥T-TN,h∥0,Ω→0(N→∞,h→0). Taking u=EuN,h and by virtue of R(TN,h,z)uN,h=(λN,h-1-z)-1uN,h, Lemma 6 and (22), we have
(55)∥u-uN,h∥0,Ω=∥EuN,h-EN,huN,h∥0,Ω≤C∥(T-TN,h)uN,h∥0,Ω≤C(∥(T-TN,h)u∥0,Ω+∥(T-TN,h)(uN,h-u)∥0,Ω),
from which follows
(56)∥u-uN,h∥0,Ω≤C∥(T-TN,h)u∥0,Ω∥u-uN,h∥0,Ω≤Chr2+τ1-1∥u∥t1,ΩNr2+t1-1,
which is (52). By direct calculation, we have
(57)∥u-uN,h∥1,Ω=∥λTu-λhTN,huN,h∥1,Ω≤∥λTu-λTN,hu∥1,Ω+∥λTN,hu-λhTN,huN,h∥1,Ω≤∥(T-TN,h)(λu)∥1,Ω+C∥λu-λN,huN,h∥0,Ω.
Plugging (20), (52), and (53) into (57) yields (51).
If(λ,u) is an eigenpair of (24), let uN,h=EN,hu; by the same argument we can prove (51) and (52).


## 3.1. Spectral and Spectral-Element Approximations for Eigenvalue Problems

Consider the following eigenvalue problem corresponding to the boundary value problem (1):
(23)Lu=λu,inΩ,u=0,on∂Ω.The variational formation of (23) is given by the following: find λ∈ℂ,0≠u∈H01(Ω), such that
(24)a(u,v)=λ(u,v),∀v∈H01(Ω).The spectral approximation scheme of (24) is given by the following: find λN∈ℂ,0≠uN∈SN(Ω), such that
(25)a(uN,vN)=λN(uN,vN),∀vN∈SN(Ω).The spectral element approximation scheme of (24) is given by the following: find λN,h∈ℂ,0≠uN,h∈SN,h(Ω), such that
(26)a(uN,h,vN,h)=λN,h(uN,h,vN,h),∀vN,h∈SN,h(Ω),Define the solution operatorsT:L2(Ω)→H01(Ω),TN:L2(Ω)→SN(Ω), and TN,h:L2(Ω)→SN,h(Ω) by
(27)a(Tf,v)=(f,v),∀f∈L2(Ω),∀v∈H01(Ω),cccicia(TNf,vN)=(f,vN),∀f∈L2(Ω),ffffffffffffffffffffffffffgffffff∀vN∈SN(Ω),22iiia(TN,hf,vN,h)=(f,vN,h),∀f∈L2(Ω),ddddddddddgddffffdd∀vN,h∈SN,h(Ω).
Obviously (see [17]), the equivalent operator forms for (24) and (26) are the following.Findλ∈ℂ,0≠u∈H01(Ω), such that
(28)Tu=λ-1u.FindλN,h∈ℂ,0≠uN,h∈SN,h(Ω), such that
(29)TN,huN,h=λN,h-1uN,h.
The adjoint problem of the eigenvalue problem (23) is
(30)L*u*=λ*u*,inΩ,u*=0,on∂Ω,
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*∈H01(Ω), such that
(31)a*(u*,v):=a(v,u*)¯=λ*(u*,v),∀v∈H01(Ω).The spectral element approximation scheme of (31) is given by the following: find λN,h*∈ℂ,0≠uN,h*∈SN,h(Ω), such that
(32)a*(uN,h*,vN,h)=λN,h*(uN,h*,vN,h),∀vN,h∈SN,h(Ω).We can likewise define the equivalent operator forms for the eigenvalue problems (31) and (32) as
(33)T*u*=λ*-1u*,TN,h*uN,h*=λN,h*-1uN,h*.Letλ 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 λN,h, λ* and λN,h* can be defined.Letλ be an eigenvalue of (23) with the algebraic multiplicity q and the ascent μ. Assume ∥TN,h-T∥1,Ω→0(N→∞,h→0); then there are eigenvalues λj,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 MN,h(λ) be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of TN,h that converge to λ.In view of adjoint problems (31) and (32), the definitions of M*(λ*) and MN,h*(λ*) are analogous to M(λ) and MN,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 whenb=0, both (24) and (26) are symmetric. Thus, the ascent μ=1 of λ, and the ascent l=1 of λN,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 finenessh not considered.Assume thatR and U are two closed subspace in H01(Ω).Denote(34)δ(R,U)=supv∈R∥v∥1,Ω=1dist(v,U),θ(R,U)=max(δ(R,U),δ(U,R)).
We say that θ(R,U) is the gap between R and U.Denote(35)εN,h=εN,h(λ)=supu∈M^(λ)infv∈SN,h(Ω)∥u-v∥1,Ω,εN,h*=εN,h*(λ*)=supu∈M*^(λ*)infv∈SN,h(Ω)∥u-v∥1,Ω.We give the following four lemmas from Theorem 8.1–8.4 in [17], which are applications to spectral element methods.Lemma 1.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0). For small enough h and big enough N, there holds
(36)θ(M(λ),MN,h(λ))≤CεN,h.Lemma 2.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0); then
(37)|λ-1-1q∑j=1q‍λj,N,h-1|≤CεN,hεN,h*.Lemma 3.
Assume that∥TN,h-T∥1,Ω→0(N→∞,h→0); then there holds
(38)|λ-λj,N,h|≤C(εN,hεN,h*)1/μ(j=1,2,…,q).
Since ker((λ-1-T)l)(l≥1) is a finite dimensional space, there exists a direct-sum decomposition H01(Ω)=ker((λ-1-T)l)⊕Ml. We define the operator El as a projection along Ml from H01(Ω) to ker((λ-1-T)l).Lemma 4.
Assume∥TN,h-T∥1,Ω→0(N→∞,h→0). Let λN,h be an eigenvalue of TN,h and limN→∞,h→0λN,h=λ. uN,h satisfies (λN,h-1-TN,h)kuN,h=0 and ∥uN,h∥1,Ω=1, where k≤μ is a positive integer. Then, for every integer l∈[k,μ], there holds
(39)∥uN,h-EluN,h∥1,Ω≤CεN,h(l-k+1)/μ.
We assume that in this section, for the sake of simplicity,Nκ=N,∀κ∈Kh.Theorem 5.
IfM(λ)⊂Ht1(Ω) and M*(λ*)⊂Ht2(Ω), then there holds the following error estimates:
(40)|1q∑j=1q‍λj,N,h-λ|≤C(hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω,(41)|λj,N,h-λ|≤C((hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω)1/μdddddddddddddddddddddddd(j=1,2,…,q),(42)θ(M(λ),MN,h(λ))≤Chτ1-1Nt1-1supu∈M^(λ)∥u∥t1,Ω
Let ∥uN,h∥1,Ω=1, and let (λN,h-1-TN,h)l1uN,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
(43)∥uN,h-u′∥1,Ω≤C((hτ1-1Nt1-1)supu∈M^(λ)∥u∥t1,Ω)(l2-l1+1)/μ,
where τ1=min(N+1,t1), τ2=min(N+1,t2).Proof.
We derive from the error estimate (20) that
(44)∥TN,h-T∥1,Ω=supf∈H01(Ω)∥(T-TN,h)f∥1,Ω∥f∥1,Ω≤C(1+r1)hr1N-r1⟶0(N⟶∞,h⟶0).
By (14),
(45)εN,h=εN,h(λ)=supu∈M^(λ)infv∈SN,h(Ω)∥u-v∥1,Ω≤C(hτ1-1Nt1-1)supu∈M^(λ)∥u∥t1,Ω.
Analogically,
(46)εN,h*≤C(hτ2-1Nt2-1)supu∈M*^(λ*)∥u∥t2,Ω.
Plugging the two inequalities above into (36), (38), and (39) yields (42), (41), and (43), respectively. We find from (37) that
(47)|1q∑j=1q‍λj,N,h-λ|=|1q∑j=1q‍λj,N,h-1-λ-1λ-1λj,N,h-1|≤C|1q∑j=1q‍λj,N,h-1-λ-1|≤CεN,hεN,h*,
combining with (45) and (46) yields (40).Supposing that∥TN,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(48)R(z)=(T-z)-1,R(TN,h,z)=(TN,h-z)-1.Define the spectral projection operators(49)E=-12iπ∫Γ‍R(T,z)dz:H01(Ω)⟶M(λ),EN,h=-12iπ∫Γ‍R(TN,h,z)dz:H01(Ω)⟶MN,h(λ).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∥TN,h-T∥0,Ω→0(N→∞,h→0), then there holds that EN,h→E(p), R(TN,h,z) is uniformly bounded with N and h, and
(50)∥(EN,h-E)v∥0,Ω≤Cmaxz∈Γ∥(T-TN,h)R(z)v∥0,Ω,fffffffffffffddddddddgddff∀v∈H01(Ω),∥(EN,h-E)v∥0,Ω≤Cmaxz∈Γ∥(T-TN,h)R(TN,h,z)v∥0,Ω,ddddddddddddddddffffffffgfffffffffffffdddd∀v∈H01(Ω).Theorem 7.
Under the assumptions of Theorem5, further assume that the ascent of λ is μ=1. Let (λN,h,uN,h) be an eigenpair of (26) with ∥uN,h∥0,Ω=1; then there exists an eigenpair (λ,u) of (24), such that
(51)∥uN,h-u∥1,Ω≤Chτ1-1∥u∥t1,ΩNt1-1,(52)∥uN,h-u∥0,Ω≤Chr2+τ1-1∥u∥t1,ΩNr2+t1-1,(53)|λN,h-λ|≤C((hτ1+τ2-2Nt1+t2-2)supu∈M^(λ)∥u∥t1,Ωsupv∈M*^(λ*)∥v∥t2,Ω),
where τ1=min(N+1,t1) and τ2=min(N+1,t2).
Let(λ,u) be an eigenpair of (24). If λN,h is an eigenvalue of (26) convergence to λ, then there exists uN,h∈ker(λN,h-1-TN,h), such that (51)–(53) hold.Proof.
We deduce (53) immediately from (41). We derive from (22) and (7) that
(54)∥Tf-TN,hf∥0,Ω≤CN-r1-r2hr1+r2∥f∥0,Ω;
thus,∥T-TN,h∥0,Ω→0(N→∞,h→0). Taking u=EuN,h and by virtue of R(TN,h,z)uN,h=(λN,h-1-z)-1uN,h, Lemma 6 and (22), we have
(55)∥u-uN,h∥0,Ω=∥EuN,h-EN,huN,h∥0,Ω≤C∥(T-TN,h)uN,h∥0,Ω≤C(∥(T-TN,h)u∥0,Ω+∥(T-TN,h)(uN,h-u)∥0,Ω),
from which follows
(56)∥u-uN,h∥0,Ω≤C∥(T-TN,h)u∥0,Ω∥u-uN,h∥0,Ω≤Chr2+τ1-1∥u∥t1,ΩNr2+t1-1,
which is (52). By direct calculation, we have
(57)∥u-uN,h∥1,Ω=∥λTu-λhTN,huN,h∥1,Ω≤∥λTu-λTN,hu∥1,Ω+∥λTN,hu-λhTN,huN,h∥1,Ω≤∥(T-TN,h)(λu)∥1,Ω+C∥λu-λN,huN,h∥0,Ω.
Plugging (20), (52), and (53) into (57) yields (51).
If(λ,u) is an eigenpair of (24), let uN,h=EN,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 Kh 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′∈Kh, κ¯∩κ′¯≠∅,
(58)γ-1hκ≤hκ′≤γhκ,γ-1(Nκ+1)≤Nκ′+1≤γ(Nκ+1).We refer to thehp-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 partitionKh 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:H01(Ω)→SN,h(Ω), such that
(59)∥v-Iv∥0,κ≤ChκNκ∥∇v∥0,ωκ,(60)∥v-Iv∥0,e≤CheNe∥∇v∥0,ωe,
where he is the length of the edge e and Ne=max(Nκ1,Nκ2), where κ1,κ2 are elements sharing the edge e and ωκ,ωe are patches covering κ and e with a few layers, respectively.
Define intervalI^=(0,1) and weight function ΦI^(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 πN of degree N,
(61)∫I^‍ΦI^α(x)|πN(x)|2dx≤CN2(β-α)∫I^‍ΦI^β|πN(x)|2dx.Lemma 10.
Let-1<α<β,σ∈[0,1]. Then there exist C1=C(α,β),C2=Cσ>0, such that for all N∈ℕ and all polynomials πN of degree bi-N,
(62)∫κ^‍Φκ^α|πN|2dxdy≤C1N2(β-α)∫κ^‍Φκ^β|πN|2dxdy,(63)∫κ^‍Φκ^2σ|∇πN|2dxdy≤C2N2(2-σ)∫κ^‍Φκ^σ|πN|2dxdy.Lemma 11.
Letα∈(1/2,1].e^:=(0,1)×{0}, Φe^:=x(1-x); then there exists Cα>0 such that for all N∈ℕ,ε∈(0,1], and all univariate polynomials π of degree N, there exists an extension ve^∈H1(κ^) and holds
(64)ve^|e^=π·Φe^α,ve^|∂κ^∖e^=0,(65)∥ve^∥0,κ^2≤Cαε∥πΦe^α/2∥0,e^2,(66)∥∇ve^∥0,κ^2≤Cα(εN2(2-α)+ε-1)∥πΦe^α/2∥0,e^2.It is easy to know that the three lemmas above hold for complex-valued polynomials.LetDκ,bκ, and cκ be the interpolations of D, b, and c in κ with the polynomial degree Nκ (resp. degree Nκ in every direction), respectively, or the L2(κ)-projection on the space of polynomials with degree Nκ. 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. (λN,h,uN,h) and (λN,h*=λN,h¯,uN,h*) are the solutions of the corresponding spectral element approximations (26) and (32), respectively.Denote(67)LκuN,h:=-∇·(Dκ∇uN,h)+bκ·∇uN,h+cκuN,h,Lκ*uN,h*:=-∇·(Dκ∇uN,h*)-bκ·∇uN,h*+(cκ-∇·bκ)uN,h*.
Define the local error indicators
(68)ηα;κ2:=ηα;Bκ2+ηα;Eκ2,ηα;κ*2:=ηα;Bκ*2+ηα;Eκ*2.
Their first terms ηα;Bκ2,ηα;Bκ*2 are the weighted element internal residuals given by
(69)ηα;Bκ2:=hκ2Nκ2∥(-LκuN,h+λN,huN,h)Φκα/2∥0,κ2,ηα;Bκ*2:=hκ2Nκ2∥(-Lκ*uN,h*+λN,h*uN,h*)Φκα/2∥0,κ2.
Their second terms ηα;Eκ2,ηα;Eκ*2 are the weighted element boundary residuals given by
(70)ηα;Eκ2:=∑e⊂∂κ∩Ω‍he2Ne∥Dκ[∂uN,h∂n]Φeα/2∥0,e2,ηα;Eκ*2:=∑e⊂∂κ∩Ω‍he2Ne∥Dκ[∂uN,h*∂n]Φeα/2∥0,e2,
where we denote the jump of the normal derivatives of uN,h and uN,h* across the edges by [∂uN,h/∂n] and [∂uN,h*/∂n], respectively. he is the length of edge e. The weight functions Φκ and Φe are scaled transformations of the weight functions  Φκ^ and Φe^; that is, if Fκ is the element map for element κ and e is the image of the edge e^ under Fκ, then
(71)Φκ=CκΦκ^∘Fκ-1,Φe=CeΦe^∘Fκ-1,
where we choose Cκ,Ce>0, such that
(72)∫κ‍Φκdxdy=∫κ‍dxdy,∫e‍Φeds=∫e‍ds.
We define the global error indicators as follows:
(73)ηα2:=∑κ∈Kh‍ηα;κ2,ηα*2:=∑κ∈Kh‍ηα;κ*2.Theorem 12.
Letα∈[0,1]. Then there exists a constant C>0 independent of h,N, and κ, such that
(74)∥u-uN,h∥1,Ω2≤C∑κ∈Kh‍Nκ2αηα;κ2+C∑κ∈Kh‍{heNe∥D-Dκ∥0,e2∥∂uN,h∂n∥0,∞,e2hκ2Nκ2∥LκuN,h-LuN,h∥0,κ2ffffffffffff+∑e⊂∂κ∩Ω‍heNe∥D-Dκ∥0,e2fffffffffrrrrrrrrrrfff×∥∂uN,h∂n∥0,∞,e2}+C∥λu-λN,huN,h∥0,Ω2.Proof.
We denotew:=u-uN,h-I(u-uN,h), where I is hp-clément operator given by Lemma 8. We derive from H01(Ω)-elliptic of a(·,·) that
(75)C∥u-uN,h∥1,Ω2≤a(u-uN,h,w)+a(u-uN,h,I(u-uN,h))=λ∫Ω‍uw¯-a(uN,h,w)ddd+∫Ω‍(λu-λN,huN,h)dddddd×I(u-uN,h)¯=∫Ω‍(λN,huN,h)w¯+∫Ω‍(λu-λN,huN,h)u-uN,h¯gggg-a(uN,h,w),a(uN,h,w)=∑κ∈κh‍∫κ‍LuN,hw¯-∑κ∈κh‍∫∂κ‍D∂uN,h∂nw¯=∑κ∈κh‍∫κ‍LuN,hw¯-12∑κ∈κh‍∑e⊂∂κ∩Ω‍∫e‍D[∂uN,h∂n]w¯.
Therefore,
(76)C∥u-uN,h∥1,Ω2≤∑κ∈Kh‍∫κ‍(-LuN,h+λN,huN,h)w¯ssssssss+12∑κ∈Kh‍∑e⊂∂κ∩Ω‍∫e‍D[∂uN,h∂n]w¯+∫Ω‍(λu-λN,huN,h)(u-uN,h)¯,
which together with
(77)∫e‍D[∂uN,h∂n]w¯=∫e‍(D-Dκ)[∂uN,h∂n]w¯+∫e‍Dκ[∂uN,h∂n]w¯,∫κ‍(-LuN,h+λN,huN,h)w¯=∫κ‍(-LκuN,h+λN,huN,h)w¯+∫κ‍(LκuN,h-LuN,h)w¯
and using Cauchy-Schwartz inequality, the hp-clément interpolation estimates in Lemma 8 then yield
(78)∥u-uN,h∥1,Ω2≤C{∑κ∈Kh‍[∑e⊂∂κ∩ΩheNe∥(D-Dκ)[∂uN,h∂n]∥0,e2η0;Bκ2+η0;Eκ2+hκ2Nκ2∥LκuN,h-LuN,h∥0,κ2sssssssssssss+∑e⊂∂κ∩ΩheNe∥(D-Dκ)[∂uN,h∂n]∥0,e2]}1/2×∥u-uN,h∥1,Ω+C∥λu-λN,huN,h∥0,Ω∥u-uN,h∥0,Ω.
Using scaled transformation and settingα=0,β=α in (61) and (62), we get  η0;Eκ≤CNκαηα;Eκ and η0;Bκ≤CNκαηα;Bκ; then this proof concludes.For the adjoint eigenvalue problem, we still have the following.Theorem 13.
Letα∈[0,1]. Then there exists a constant C>0 independent of h,N, and κ, such that
(79)∥u*-uN,h*∥1,Ω2≤C∑κ∈Kh‍Nκ2αηα;κ*2+C∑κ∈Kh‍{∥∂uN,h*∂n∥0,∞,e2hκ2Nκ2∥Lκ*uN,h*-L*uN,h*∥2cccccccccchkkhcc+∑e⊂∂κ∩Ω‍heNe∥D-Dκ∥0,e2∥∂uN,h*∂n∥0,∞,e2}+C∥λ*u*-λN,h*uN,h*∥0,Ω2.Lemma 14.
Letα∈[0,1],ε>0. Then there exists a constant C(ε)>0 independent of h,N, and κ, such that
(80)ηα;Bκ2≤C(ε){Nκ2(1-α)∥u-uN,h∥1,κ2ffffffffffiiff+Nκmax{1+2ε-2α,0}hκ2Nκ2ffffffffffiiff×(∥λN,huN,h-λu+LuN,h-LκuN,h∥0,k2)}.Proof.
We denotevκ:=(-LκuN,h+λN,huN,h)Φκα∈H01(κ) with α∈(0,1] and extend vκ to Ω by vκ=0 on Ω∖κ; then
(81)∥vκΦκ-α/2∥0,κ2=∫κ‍(-LκuN,h+λN,huN,h)vκ¯=-∫κ‍(LκuN,h)vκ¯+a(u,vκ)+∫κ‍(λN,huN,h-λu)vκ¯=a(u-uN,h,vκ)+∫κ(λN,huN,h-λu+LuN,h-LκuN,h)vκ¯≤C∥u-uN,h∥1,κ|vκ|1,κ+∥Φκα/2(λN,huN,h-λu555+LuN,h-LκuN,h)Φκα/2∥0,κ×∥vκΦκ-α/2∥0,κ.
We consider the H1 semi norm for vκ. 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
(82)|vκ|1,κ2≤2∫κ‍Φκ2α|∇(λN,huN,h-LκuN,h)|2+2∫κ‍|∇Φκα|2|λN,huN,h-LκuN,h|2≤CNκ2(2-α)hκ2∫κΦκα|λN,huN,h-LκuN,h|2+C1hκ2∫κ‍Φκ2(α-1)|λN,huN,h-LκuN,h|2≤CNκ2(2-α)hκ2∫κ‍Φκα|λN,huN,h-LκuN,h|2=CNκ2(1-α)Nκ2hκ2∥vκΦκ-α/2∥0,κ2.
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.
Sinceηα;Bκ=hκ/Nκ∥vκΦκ-α/2∥0,κ, we obtain
(83)ηα;Bκ≤C(+λN,huN,h-λu∥0,κNκ1-α∥u-uN,h∥1,κ222h2+hκNκ∥LuN,h-LκuN,h22hhhhhhh22+λN,huN,h-λu∥0,κ).
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
(84)Nκhκηα;Bκ=Φκα/2∥(λN,huN,h-LκuN,h)∥0,κ≤CNκβ-α∥(λN,huN,h-LκuN,h)Φκβ/2∥0,κ=CNκβ-αNκhκηβ;Bκ≤CNκβ-α(Nκ1-βNκhκ∥u-uN,h∥1,κ555555555i+∥λN,huN,h-λu+LuN,h-LκuN,h∥0,κNκhκ).
Setting β=1/2+ε,ε>0,
(85)ηα;Bκ≤C(ε){Nκ1-α∥u-uN,h∥1,κfffffffff+Nκ1/2+ε-αhκNκfffff×∥λN,huN,h-λu+LuN,h-LκuN,h∥0,κ}.
We obtain the desired result immediately from (83) and (85).In order to obtain a local upper bound for the error indicatorηα;κ, we consider the edge residual term  ηα;Eκ. we introduce the set
(86)ωκ:=∪{κ′|κ′andκshareatleastoneedge}.Lemma 15.
Letα∈[0,1],ε>0. Then there exists a constant C(ε)>0 independent of h,N, and κ, such that
(87)ηα;Eκ2≤C(ε)Nκmax(1-2α+2ε,0)×{Nκ∥u-uN,h∥1,ωκ2+Nκ2εhκ2Nκ2hhh×∥λN,huN,h-λu+LuN,h-LκuN,h∥0,ωκ2Nκ2εhκ2Nκ2}.Proof.
We will use weight functions on edge and a suitable extension operator. For a given elementκ with edge e, we choose the element κ1 so that ∂κ1∩∂κ=e. Denote κe¯:=κ1¯∪κ¯; we construct a function  we∈H01(κe) with we|e=Dκ[∂uN,h/∂n]Φeα as follows.
Letve^=CeDκ[∂uN,h/∂n]Φe^α (Ce is defined by (71)). Using Lemma 11, we extend ve^ to κ^, where the polynomial π corresponds to CeDκ[∂uN,h/∂n]. Define  we|κ and we|κ1 as the affine transformation of ve^ in κ^; Thus, we is a piecewise H1-function. From (64), we know we vanishes on  ∂κe; Therefore,  we∈H01(κe). It is trivial to extend we to Ω, such that we=0 in Ω∖κe. We find
(88)∥Dκ[∂uN,h∂n]Φeα/2∥0,e2=∫e‍D[∂uN,h∂n]we¯+∫e‍(Dκ-D)[∂uN,h∂n]we¯=∫κe‍LuN,hwe¯-a(uN,h,we)+∫e‍(Dκ-D)[∂uN,h∂n]we¯=∫κe‍(LuN,h-λu)we¯+a(u-uN,h,we)+∫e‍(Dκ-D)[∂uN,h∂n]we¯≤∥LuN,h-λu∥0,κe∥we∥0,κe+C∥u-uN,h∥1,κe|we|1,κe+∥(Dκ-D)Dκ∥0,∞,e∥Dκ[∂uN,h∂n]Φeα/2∥0,e2.
Therefore,
(89)∥Dκ[∂uN,h∂n]Φeα/2∥0,e2≤C∥LuN,h-λu∥0,κe∥we∥0,κe+C∥u-uN,h∥1,κe|we|1,κe.
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 ∥we∥0,κe and |we|1,κe as follows:
(90)|we|1,κe2≤C1hκ(εNκ2(2-α)+ε-1)×∥Dκ[∂uN,h∂n]Φeα/2∥0,e2,∥we∥0,κe≤Chκε∥Dκ[∂uN,h∂n]Φeα/2∥0,e2.
It follows from (89)-(90) that
(91)∥Dκ[∂uN,h∂n]Φeα/2∥0,e≤C{(1hκ(εNκ2(2-α)+ε-1))1/2gggggg×∥u-uN,h∥1,κe+(hκε)1/2∥∥LuN,h-λu∥0,κe(1hκ(εNκ2(2-α)+ε-1))}.
By the definition of ηα;Eκ2 and setting α=0 in Lemma 14, we get
(92)∥LκuN,h-λN,huN,h∥0,k2≤C(ε){Nκ4hκ2∥u-uN,h∥1,k2+Nκ1+2εdddddddid×∥λN,huN,h-λu+LuN,h-LκuN,h∥0,k2Nκ4hκ2},
by the triangle inequality
(93)∥LuN,h-λu∥0,κe≤∥LκuN,h-λN,huN,h∥0,κe+∥λN,huN,h-λu+LuN,h-LκuN,h∥0,κe.
Combining the three inequalities above and summing, we have
(94)ηα;Eκ2≤C{1Nκ(εNκ2(2-α)+ε-1)+Nκ3ε}×∥u-uN,h∥1,ωκ2+CεNκ2(1+ε)hκ2Nκ2×∥λN,huN,h-λu+LuN,h-LκuN,h∥0,ωκ2.
Settingε=1/Nκ2 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 ηα;Eκ≤Nκβ-αηβ;Eκ, and find the desired result.Combining Lemmas14 and 15, we obtain the following theorem.Theorem 16.
Letα∈[0,1],ε>0. Then there exists a constant C>0 independent of h,N, and κ, such that
(95)ηα;κ2≤C(ε)Nκmax(1-2α+2ε,0)×{Nκ∥u-uN,h∥1,ωκ2+Nκ2εhκ2Nκ2xxx×∥λN,huN,h-λu+LuN,h-LκuN,h∥0,ωκ2hκ2Nκ2}.Similarly, we have Theorem17.Theorem 17.
Letα∈[0,1],ε>0. Then there exists a constant C>0 independent of h,N, and κ, such that
(96)ηα;κ*2≤C(ε)Nκmax(1-2α+2ε,0)×{Nκ∥u*-uN,h*∥1,ωκ2+Nκ2εhκ2Nκ2xxx×∥λN,h*uN,h*-λ*u*+L*uN,h*-Lκ*uN,h*∥0,ωκ2hκ2Nκ2}.In order to estimate bounds of|λ-λN,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*∈H01(Ω), (w,w*)≠0,
(97)a(w,w*)(w,w*)-λ=a(w-u,w*-u*)(w,w*)-λ(w-u,w*-u*)(w,w*).Theorem 19.
Under the assumptions of Theorem7, 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
(98)∥u-uN,h∥1,Ω≤C(∑κ∈Kh‍Nκ2αηα;κ2)1/2,(99)ηα;κ2≤C(ε)Nκmax(2-2α+2ε,1)∥u-uN,h∥1,ωκ2.
Further let the ascent of λN,h be l=1, and let (λN,h*,uN,h*) be the corresponding adjoint eigenpair of (32), then there exists an adjoint eigenpair (λ*,u*) of (31), such that
(100)|λN,h-λ|≤C(∑κ∈Kh‍Nκ2αηα;κ2)1/2(∑κ∈Kh‍Nκ2αηα;κ*2)1/2.
Particularly, if the eigenvalue problem (23) is symmetric (i.e., b=0), then
(101)CC(ε)-1∑κ∈Kh‍Nκmin(2α-2-2ε,-1)ηα;κ2≤|λN,h-λ|.Proof.
We know from the assumptionD,c∈Ht1(κ),b∈(Ht1(κ))2. By the interpolation error estimates (14) and (15), we have
(102)∥LκuN,h-LuN,h∥0,κ≤Chκmin(Nκ+1,t1)-1Nκ-t1+1.
From D∈Ht1(κ), we know that D∈Ht1-1/2(e). By the interpolation error estimate on edge of element (see formula (5.4.42) in [2]), we get
(103)∥D-Dκ∥0,e≤Chemin(Nκ+1,t1-1/2)Nκ-t1+1/2.
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=Nκ, that
(104)∥λu-λN,huN,h∥0,Ω≤|λ-λN,h|∥u∥0,Ω+|λN,h|∥u-uN,h∥0,Ω≤Chτ1+τ2-2Nt1+t2-2+Chr2+τ1-1Nr2+t1-1≤Chr2+τ1-1Nr2+t1-1.
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 (uN,h,uN,h*)=1 and uN,h* is uniformly bounded with h and N. By the same argument of (98), we can deduce that
(105)∥u*-uN,h*∥1,Ω2≤C∑κ∈Kh‍Nκ2αηα;κ*2.
From (97), we have
(106)a(uN,h,uN,h*)(uN,h,uN,h*)-λ=a(uN,h-u,uN,h*-u*)(uN,h,uN,h*)-λ(uN,h-u,uN,h*-u*)(uN,h,uN,h*);
that is,
(107)λN,h-λ=a(uN,h-u,uN,h*-u*)-λ(uN,h-u,uN,h*-u*).
Substituting (98) and (105) into the above equality, we obtain (100).
If the eigenvalue problem (23) is symmetric (i.e., b=0), then
(108)λN,h-λ=a(uN,h-u,uN,h-u)-λ(uN,h-u,uN,h-u).
Up to higher order term λ(uN,h-u,uN,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 compute1/|(uN,h,uN,h*)| as condition number for simple eigenvalue (see Remark 2.1 in [25]), where uN,h and uN,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(109)-Δu+10ux+uy=λu,inΩ=(0,1)2,u=0,on∂Ω.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

Figure1 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.
N
DOF
LGL-SM
Modal-SM
Eq-SM
λ
1
λ
1
λ
1
4
9
5.19
E
+
00
5.19
E
+
00
5.19
E
+
00
5
16
4.51
E
-
01
4.51
E
-
01
4.51
E
-
01
6
25
7.68
E
-
03
7.68
E
-
03
7.68
E
-
03
7
36
1.07
E
-
05
1.07
E
-
05
1.07
E
-
05
8
49
1.21
E
-
05
1.21
E
-
05
1.21
E
-
05
9
64
9.16
E
-
07
9.16
E
-
07
9.16
E
-
07
10
81
2.46
E
-
08
2.46
E
-
08
2.48
E
-
08
11
100
2.91
E
-
10
2.91
E
-
10
4.35
E
-
09
12
121
9.31
E
-
13
1.06
E
-
12
2.79
E
-
08
13
144
5.68
E
-
14
1.28
E
-
13
1.41
E
-
07
14
169
2.84
E
-
14
1.28
E
-
13
2.28
E
-
06
15
196
7.82
E
-
14
2.13
E
-
14
3.60
E
-
05Figure 1
Condition number of first eigenvalue for SM.


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

Tables1 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
Error
DOF
Error
DOF
Error
DOF
Error
DOF
2
5.18
E
+
00
9
2.54
E
-
01
49
1.50
E
-
02
225
9.00
E
-
04
961
3
7.00
E
-
03
25
6.10
E
-
04
121
1.20
E
-
05
529
1.90
E
-
07
2209
4
8.40
E
-
03
49
2.60
E
-
05
225
9.70
E
-
08
961
3.70
E
-
10
3969
5
1.64
E
-
04
81
1.60
E
-
07
361
1.50
E
-
10
1521
1.30
E
-
13
6241
6
4.10
E
-
07
121
2.30
E
-
11
529
9.90
E
-
13
2209
3.60
E
-
12
9025
7
3.10
E
-
08
169
1.70
E
-
12
729
3.10
E
-
13
3025
1.60
E
-
12
12321
8
1.90
E
-
10
225
1.90
E
-
13
961
2.10
E
-
12
3969
4.80
E
-
12
16129
9
5.50
E
-
13
289
2.80
E
-
14
1225
6.00
E
-
13
5041
1.10
E
-
12
20449
10
3.80
E
-
13
361
1.10
E
-
12
1521
4.40
E
-
12
6241
1.50
E
-
11
25281In 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 Table4, 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
7.00
E
-
03
6.10
E
-
04
1.20
E
-
05
1.90
E
-
07
4
8.40
E
-
03
2.60
E
-
05
9.70
E
-
08
3.70
E
-
10
5
1.60
E
-
04
1.60
E
-
07
1.50
E
-
10
1.30
E
-
12
6
4.10
E
-
07
2.40
E
-
11
3.60
E
-
13
8.60
E
-
12
7
3.10
E
-
08
6.10
E
-
12
1.30
E
-
11
3.00
E
-
11
8
1.80
E
-
10
3.10
E
-
11
2.30
E
-
10
2.10
E
-
10
9
7.50
E
-
11
3.40
E
-
11
6.80
E
-
10
7.40
E
-
10
10
2.50
E
-
11
9.90
E
-
10
8.70
E
-
09
6.60
E
-
09
11
2.00
E
-
09
9.60
E
-
09
8.90
E
-
09
5.40
E
-
07Table 4
Condition number of first eigenvalue for hp-SEM.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
4.284381324
4.270132842
4.269625046
4.269615821
4
4.267343095
4.269607452
4.269615638
4.26961567
5
4.269636446
4.269615725
4.26961567
4.26961567
6
4.269619135
4.26961567
4.26961567
4.26961567
7
4.269615617
4.26961567
4.26961567
4.26961567
8
4.26961567
4.26961567
4.26961567
4.26961567
9
4.26961567
4.26961567
4.26961567
4.26961567Remark 21.
Condition numbers of 1st eigenvalue for hp-FEM (not listed in Table4) are almost the same to those for hp-SEM.


#### 5.1.4. Validity of the Error Indicator

Denote(110)ψα=(∑κ∈Kh‍Nκ2αηα;κ2)1/2(∑κ∈Kh‍Nκ2αηα;κ*2)1/2.From Theorem19, we know that ψα is a reliable error indicator for λN,h. We choose ψ0 (setting α=0 in (110)) as a posteriorii error indicator.In Figures2 and 3, we denote the true error and est. error with |λN,h-λ| and ψ0, respectively.Figure 2
The Error indicatorψ0 of LGL-SM.Figure 3
The Error indicatorψ0 of hp-SEM (h=1/2).As is depicted in Figure2, 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(111)-Δu+10ux=λu,inΩ=(-1,1)2(0,1)2,u=0,on∂Ω.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 indicatorψ0 of P-SEM.
N
λ
N
,
h
ψ
0
3
28.56900
2.72
E
+
01
4
31.99175
3.49
E
+
00
5
34.82082
2.25
E
-
01
6
34.65087
1.31
E
-
02
7
34.65057
3.32
E
-
03
8
34.64765
1.92
E
-
03
9
34.64567
1.22
E
-
03
10
34.64432
8.11
E
-
04
11
34.64335
5.62
E
-
04
12
34.64265
4.02
E
-
04
13
34.64212
2.95
E
-
04
14
34.64171
2.22
E
-
04
15
34.64139
1.71
E
-
04
16
34.64114
1.33
E
-
04
17
34.64094
1.06
E
-
04
18
34.64078
8.49
E
-
05


#### 5.2.1. Stability of P-Version Methods

Figure4 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 finenessh=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.


## 5.1. Example 1

Consider the nonsymmetric eigenvalue problem(109)-Δu+10ux+uy=λu,inΩ=(0,1)2,u=0,on∂Ω.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

Figure1 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.
N
DOF
LGL-SM
Modal-SM
Eq-SM
λ
1
λ
1
λ
1
4
9
5.19
E
+
00
5.19
E
+
00
5.19
E
+
00
5
16
4.51
E
-
01
4.51
E
-
01
4.51
E
-
01
6
25
7.68
E
-
03
7.68
E
-
03
7.68
E
-
03
7
36
1.07
E
-
05
1.07
E
-
05
1.07
E
-
05
8
49
1.21
E
-
05
1.21
E
-
05
1.21
E
-
05
9
64
9.16
E
-
07
9.16
E
-
07
9.16
E
-
07
10
81
2.46
E
-
08
2.46
E
-
08
2.48
E
-
08
11
100
2.91
E
-
10
2.91
E
-
10
4.35
E
-
09
12
121
9.31
E
-
13
1.06
E
-
12
2.79
E
-
08
13
144
5.68
E
-
14
1.28
E
-
13
1.41
E
-
07
14
169
2.84
E
-
14
1.28
E
-
13
2.28
E
-
06
15
196
7.82
E
-
14
2.13
E
-
14
3.60
E
-
05Figure 1
Condition number of first eigenvalue for SM.


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

Tables1 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
Error
DOF
Error
DOF
Error
DOF
Error
DOF
2
5.18
E
+
00
9
2.54
E
-
01
49
1.50
E
-
02
225
9.00
E
-
04
961
3
7.00
E
-
03
25
6.10
E
-
04
121
1.20
E
-
05
529
1.90
E
-
07
2209
4
8.40
E
-
03
49
2.60
E
-
05
225
9.70
E
-
08
961
3.70
E
-
10
3969
5
1.64
E
-
04
81
1.60
E
-
07
361
1.50
E
-
10
1521
1.30
E
-
13
6241
6
4.10
E
-
07
121
2.30
E
-
11
529
9.90
E
-
13
2209
3.60
E
-
12
9025
7
3.10
E
-
08
169
1.70
E
-
12
729
3.10
E
-
13
3025
1.60
E
-
12
12321
8
1.90
E
-
10
225
1.90
E
-
13
961
2.10
E
-
12
3969
4.80
E
-
12
16129
9
5.50
E
-
13
289
2.80
E
-
14
1225
6.00
E
-
13
5041
1.10
E
-
12
20449
10
3.80
E
-
13
361
1.10
E
-
12
1521
4.40
E
-
12
6241
1.50
E
-
11
25281In 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 Table4, 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
7.00
E
-
03
6.10
E
-
04
1.20
E
-
05
1.90
E
-
07
4
8.40
E
-
03
2.60
E
-
05
9.70
E
-
08
3.70
E
-
10
5
1.60
E
-
04
1.60
E
-
07
1.50
E
-
10
1.30
E
-
12
6
4.10
E
-
07
2.40
E
-
11
3.60
E
-
13
8.60
E
-
12
7
3.10
E
-
08
6.10
E
-
12
1.30
E
-
11
3.00
E
-
11
8
1.80
E
-
10
3.10
E
-
11
2.30
E
-
10
2.10
E
-
10
9
7.50
E
-
11
3.40
E
-
11
6.80
E
-
10
7.40
E
-
10
10
2.50
E
-
11
9.90
E
-
10
8.70
E
-
09
6.60
E
-
09
11
2.00
E
-
09
9.60
E
-
09
8.90
E
-
09
5.40
E
-
07Table 4
Condition number of first eigenvalue for hp-SEM.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
4.284381324
4.270132842
4.269625046
4.269615821
4
4.267343095
4.269607452
4.269615638
4.26961567
5
4.269636446
4.269615725
4.26961567
4.26961567
6
4.269619135
4.26961567
4.26961567
4.26961567
7
4.269615617
4.26961567
4.26961567
4.26961567
8
4.26961567
4.26961567
4.26961567
4.26961567
9
4.26961567
4.26961567
4.26961567
4.26961567Remark 21.
Condition numbers of 1st eigenvalue for hp-FEM (not listed in Table4) are almost the same to those for hp-SEM.


### 5.1.4. Validity of the Error Indicator

Denote(110)ψα=(∑κ∈Kh‍Nκ2αηα;κ2)1/2(∑κ∈Kh‍Nκ2αηα;κ*2)1/2.From Theorem19, we know that ψα is a reliable error indicator for λN,h. We choose ψ0 (setting α=0 in (110)) as a posteriorii error indicator.In Figures2 and 3, we denote the true error and est. error with |λN,h-λ| and ψ0, respectively.Figure 2
The Error indicatorψ0 of LGL-SM.Figure 3
The Error indicatorψ0 of hp-SEM (h=1/2).As is depicted in Figure2, 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.


## 5.1.1. Comparisons between LGL-SM, Modal, and Eq-SM

Figure1 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.
N
DOF
LGL-SM
Modal-SM
Eq-SM
λ
1
λ
1
λ
1
4
9
5.19
E
+
00
5.19
E
+
00
5.19
E
+
00
5
16
4.51
E
-
01
4.51
E
-
01
4.51
E
-
01
6
25
7.68
E
-
03
7.68
E
-
03
7.68
E
-
03
7
36
1.07
E
-
05
1.07
E
-
05
1.07
E
-
05
8
49
1.21
E
-
05
1.21
E
-
05
1.21
E
-
05
9
64
9.16
E
-
07
9.16
E
-
07
9.16
E
-
07
10
81
2.46
E
-
08
2.46
E
-
08
2.48
E
-
08
11
100
2.91
E
-
10
2.91
E
-
10
4.35
E
-
09
12
121
9.31
E
-
13
1.06
E
-
12
2.79
E
-
08
13
144
5.68
E
-
14
1.28
E
-
13
1.41
E
-
07
14
169
2.84
E
-
14
1.28
E
-
13
2.28
E
-
06
15
196
7.82
E
-
14
2.13
E
-
14
3.60
E
-
05Figure 1
Condition number of first eigenvalue for SM.


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

Tables1 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
Error
DOF
Error
DOF
Error
DOF
Error
DOF
2
5.18
E
+
00
9
2.54
E
-
01
49
1.50
E
-
02
225
9.00
E
-
04
961
3
7.00
E
-
03
25
6.10
E
-
04
121
1.20
E
-
05
529
1.90
E
-
07
2209
4
8.40
E
-
03
49
2.60
E
-
05
225
9.70
E
-
08
961
3.70
E
-
10
3969
5
1.64
E
-
04
81
1.60
E
-
07
361
1.50
E
-
10
1521
1.30
E
-
13
6241
6
4.10
E
-
07
121
2.30
E
-
11
529
9.90
E
-
13
2209
3.60
E
-
12
9025
7
3.10
E
-
08
169
1.70
E
-
12
729
3.10
E
-
13
3025
1.60
E
-
12
12321
8
1.90
E
-
10
225
1.90
E
-
13
961
2.10
E
-
12
3969
4.80
E
-
12
16129
9
5.50
E
-
13
289
2.80
E
-
14
1225
6.00
E
-
13
5041
1.10
E
-
12
20449
10
3.80
E
-
13
361
1.10
E
-
12
1521
4.40
E
-
12
6241
1.50
E
-
11
25281In 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 Table4, 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.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
7.00
E
-
03
6.10
E
-
04
1.20
E
-
05
1.90
E
-
07
4
8.40
E
-
03
2.60
E
-
05
9.70
E
-
08
3.70
E
-
10
5
1.60
E
-
04
1.60
E
-
07
1.50
E
-
10
1.30
E
-
12
6
4.10
E
-
07
2.40
E
-
11
3.60
E
-
13
8.60
E
-
12
7
3.10
E
-
08
6.10
E
-
12
1.30
E
-
11
3.00
E
-
11
8
1.80
E
-
10
3.10
E
-
11
2.30
E
-
10
2.10
E
-
10
9
7.50
E
-
11
3.40
E
-
11
6.80
E
-
10
7.40
E
-
10
10
2.50
E
-
11
9.90
E
-
10
8.70
E
-
09
6.60
E
-
09
11
2.00
E
-
09
9.60
E
-
09
8.90
E
-
09
5.40
E
-
07Table 4
Condition number of first eigenvalue for hp-SEM.
N
h
=
1
/
2
h
=
1
/
4
h
=
1
/
8
h
=
1
/
16
3
4.284381324
4.270132842
4.269625046
4.269615821
4
4.267343095
4.269607452
4.269615638
4.26961567
5
4.269636446
4.269615725
4.26961567
4.26961567
6
4.269619135
4.26961567
4.26961567
4.26961567
7
4.269615617
4.26961567
4.26961567
4.26961567
8
4.26961567
4.26961567
4.26961567
4.26961567
9
4.26961567
4.26961567
4.26961567
4.26961567Remark 21.
Condition numbers of 1st eigenvalue for hp-FEM (not listed in Table4) are almost the same to those for hp-SEM.


## 5.1.4. Validity of the Error Indicator

Denote(110)ψα=(∑κ∈Kh‍Nκ2αηα;κ2)1/2(∑κ∈Kh‍Nκ2αηα;κ*2)1/2.From Theorem19, we know that ψα is a reliable error indicator for λN,h. We choose ψ0 (setting α=0 in (110)) as a posteriorii error indicator.In Figures2 and 3, we denote the true error and est. error with |λN,h-λ| and ψ0, respectively.Figure 2
The Error indicatorψ0 of LGL-SM.Figure 3
The Error indicatorψ0 of hp-SEM (h=1/2).As is depicted in Figure2, 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(111)-Δu+10ux=λu,inΩ=(-1,1)2(0,1)2,u=0,on∂Ω.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 indicatorψ0 of P-SEM.
N
λ
N
,
h
ψ
0
3
28.56900
2.72
E
+
01
4
31.99175
3.49
E
+
00
5
34.82082
2.25
E
-
01
6
34.65087
1.31
E
-
02
7
34.65057
3.32
E
-
03
8
34.64765
1.92
E
-
03
9
34.64567
1.22
E
-
03
10
34.64432
8.11
E
-
04
11
34.64335
5.62
E
-
04
12
34.64265
4.02
E
-
04
13
34.64212
2.95
E
-
04
14
34.64171
2.22
E
-
04
15
34.64139
1.71
E
-
04
16
34.64114
1.33
E
-
04
17
34.64094
1.06
E
-
04
18
34.64078
8.49
E
-
05


### 5.2.1. Stability of P-Version Methods

Figure4 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 finenessh=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.


## 5.2.1. Stability of P-Version Methods

Figure4 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 finenessh=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.



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