A Superconvergence result for mixed finite element approximations of the eigenvalue problem

DOI:10.1051/m2an/2011065 www.esaim-m2an.org

A SUPERCONVERGENCE RESULT FOR MIXED FINITE ELEMENT APPROXIMATIONS OF THE EIGENVALUE PROBLEM

Qun Lin

and Hehu Xie

Abstract.In this paper, we present a superconvergence result for the mixed finite element approxima- tions of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Dur´anet al.[Math. Models Methods Appl. Sci.9(1999) 1165–1178] and Gardini [ESAIM:

M2AN43(2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second or- der elliptic eigenvalue problems by general mixed finite element methods which have the commuting diagram property. Some numerical experiments are given to confirm the theoretical analysis.

Mathematics Subject Classification. 65N30, 65N25, 65L15, 65B99.

Received September 26, 2010. Revised July 22, 2011.

Published online February 3, 2012.

1. Introduction

In this paper, we are concerned with the following second order elliptic eigenvalue problem: find (p, λ) such

that ⎧

⎨

⎩

−∇ ·(A∇p) +ϕp =λρp, inΩ, B(p) = 0, on∂Ω,

Ωρp²dΩ = 1,

(1.1)

where A = (a_ij)_2×2 is a symmetric positive definite matrix with a_ij ∈ W^1,∞(Ω) for 1 ≤i, j ≤ 2, 0 ≤ ϕ ∈ W^0,∞(Ω) on ¯Ω, ρ is a bounded positive function on ¯Ω and 0 < c₀ ≤ρ ∈ W^0,∞(Ω), Ω ⊂ R² is a bounded domain with Lipschitz boundary∂Ω,∇and∇·denote the gradient and divergence operators andB(p) denotes the boundary condition which can be Dirichlet or Neumann type,i.e.,

B_D(p) =p= 0, on∂Ω, (1.2)

or

B_N(p) =n· A∇p= 0, on∂Ω. (1.3)

Keywords and phrases.Second order elliptic eigenvalue problem, mixed finite element method, superconvergence.

∗This project was supported by the National Natural Science Foundation of China (11001259, 11031006, 2011CB309703).

1 LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China.[email protected];[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2012

The mixed formulation for the eigenvalue problem (1.1) comes from computing the vibration modes of a fluid in a displacement formulations since using the displacement formulation for fluid is more convenient than using the pressure or potential as variable (see,e.g., [11]).

There are several works for the second order elliptic eigenvalue problems in the mixed formulation and their numerical methods such as Babuˇska and Osborn [1,2,21], Mercier et al. [20], Boffi et al. [5,6], Boffi [4], etc.

Osborn [21], Mercier et al.[20], Boffi et al.[5], Boffi [4] give the analysis for the eigenpair approximations by mixed/hybrid finite element methods based on the theory of compact operator (see,e.g., [9]).

In [11–13], a superconvergence result between the lowest order Raviart-Thomas mixed finite element approxi- mation and their corresponding mixed finite element projection for the Laplace eigenvalue problem withA=I, ϕ= 0 andρ= 1 in (1.1) has been proven. Their analysis is based on the equivalence between the lowest order Raviart-Thomas approximation and the non-conforming Crouzeix-Raviart approximation for Laplace eigenvalue problem with Neumann boundary condition. In this paper, we extend the superconvergence result to general second order elliptic eigenvalue problems with general mixed finite element methods. This is also an extension of the superconvergence for the second order elliptic problem by mixed finite element methods (see,e.g., [8]).

The outline of the paper goes as follows. In Section 2, we introduce some preliminaries and notations, and state the weak form of the eigenvalue problem and its corresponding discrete form. A superconvergence result between the eigenfunction approximations and its corresponding interpolant is obtained in Section 3. Some numerical results are given in Section 4 to confirm the theoretical analysis in Section 3. Some concluding remarks are given in the last section.

Throughout this paperC orc, denotes a generic positive constant which is independent of the mesh size but sometimes depends on the eigenvalues of the problem (1.1).

2. Mixed finite element method

We define a new vector-valued function uas follows u=A∇p.

Then (1.1) can be transformed into the following equivalent mixed formulation

⎧⎨

⎩

−∇ ·u+ϕp=λρp, inΩ, A⁻¹ u− ∇p= 0, inΩ,

Ωρp²dΩ= 1,

(2.1)

with the boundary condition

p= 0, on∂Ω, (2.2)

for the Dirichlet boundary case (1.2) or

n·u= 0, on∂Ω, (2.3)

for the Neumann boundary case (1.3).

LetW :=L²(Ω) be the standardL²space onΩwith norm · 0 and letV be the Hilbert space V:=H(div, Ω) =

v∈(L²(Ω))² :∇ ·v∈L²(Ω)

for Dirichlet boundary condition (1.2) or V:=H₀(div, Ω) =

v∈(L²(Ω))² :∇ ·v∈L²(Ω), n·v= 0

for Neumann boundary condition (1.3) equipped with the norm vV:=

v²₀+∇ ·v²₀ ¹²,

respectively.

The corresponding weak formulation for the problem (2.1) seeks (u, p, λ)∈V×W× Rsuch thatp= 0 and

⎧⎨

⎩

a(u,v)−b(v, p) = 0, ∀v∈V, b(u, q) +d(p, q) =λr(p, q), ∀q∈W,

r(p, p) = 1,

(2.4)

wherea(·,·),b(·,·),d(·,·) andr(·,·) are bilinear forms defined by a(u,v) =

Ω

u· A⁻¹vdΩ, b(v, q) =−

Ω∇ ·vqdΩ, d(p, q) =

ΩϕpqdΩ, r(p, q) =

ΩρpqdΩ.

For the aim of the analysis, we also need to define the weightedL²(Ω) norm based on the inner productr(·,·) as

qr:=r(q, q)¹².

Based on the property of the functionρ, it is obvious that the norm · _ris equivalent to · ₀. From [2], we know that the eigenvalue problem (2.4) owns an eigenvalue sequence{λ_j}:

0≤λ₁≤λ₂≤. . .≤λ_k ≤. . . , lim

k→∞λ_k=∞, and the associated eigenfunctions

(u₁, p₁), (u₂, p₂), . . . ,(u_k, p_k), . . . , wherer(p_i, p_j) =δ_ij.

Now, let us define the mixed finite element discretization method for the eigenvalue problem (2.4). The well- posedness of the discrete weak form of (2.4) can be guaranteed by the fact that the corresponding approximation spaces satisfy the Babuˇska-Brezzi condition (see,e.g., [7,8,14]). LetT_h be a partition ofΩinto finite elements (triangles or rectangles), which is regular and has a mesh sizeh. Associated with the partitionT_h, we define the finite dimensional spacesV_h⊂VandW_h⊂W as finite element spaces [8,14].

In this paper, we assume the mixed finite element spaceV_h×W_h has an interpolation operator (I_h, J_h) I_h×J_h: V×W −→V_h×W_h

satisfying the following commuting diagram property (see,e.g., [8])

∇ ·I_hv =J_h∇ ·v, ∀v∈V∩H(Ω), (2.5)

whereJ_hdenotes theL²-projection onW_h.

The mixed finite element spaceV_h×W_his assumed to have the following approximation properties:

I_hv0≤Cv1, (2.6)

u−I_hu_V+p−J_hp₀≤Ch^m(u_m+∇ ·u_m+p_m), 0< m≤k, (2.7) for any u∈(H^m(Ω))²,∇ ·u∈H^m(Ω), and p∈H^m(Ω), wherek is a positive integer according to the degree of the mixed finite element spaceV_h×W_h.

Now, let us define the approximation of eigenpair (u, p, λ) of (2.4) by the mixed finite element method as finding a pair of (u_h, p_h, λ_h)∈V_h×W_h× Rsuch that

⎧⎨

⎩

a(u_h,v_h)−b(v_h, p_h) = 0, ∀v_h∈V_h b(u_h, q_h) +d(p_h, q_h) =λ_hr(p_h, q_h), ∀q_h∈W_h,

r(p_h, p_h) = 1.

(2.8)

In this paper, we assume the eigenfunctions of (1.1) has the regularityp∈H^1+t(Ω),i.e.,

p1+t≤C, (2.9)

where t is a positive real number and the constant C depends on the eigenfunction p. From [2] the discrete eigenvalue problem (2.8) has eigenvalues

0≤λ_1,h≤λ_2,h≤. . .≤λ_k,h ≤. . .≤λ_N,h, and the corresponding eigenfunctions

(u_1,h, p_1,h),(u_2,h, p_2,h), . . . ,(u_k,h, p_k,h), . . . ,(u_N,h, p_N,h),

where r(p_i,h, p_j,h) = δ_ij,1 ≤i, j ≤ N (N is the dimension of the mixed finite element space V_h×W_h). For simplicity, we only consider numerical approximations of simple eigenvalues in this paper.

3. A superconvergence result

In [11–13], a type of superconvergence between the eigenfunction approximation and its corresponding mixed finite element projection has been given for Laplace eigenvalue problems (A = I, ϕ = 0 and ρ = 1) by the lowest order Raviart-Thomas mixed finite element. For the general second order elliptic eigenvalue problem (1.1), there is no corresponding superconvergence result. In this section, we show the same superconvergence result for the general eigenvalue problem (1.1) with general mixed finite elements which satisfy the commuting diagram property (2.5).

In order to deduce the superconvergence result, we need to define the solution operators:T andS, and their discrete versionT_handS_h.

First, let us define the pair of solution operatorsT andS as

T :W −→W, S:W −→V such that for anyg∈W

a(Sg,v)−b(v, T g) = 0, ∀v∈V,

b(Sg, q) +d(T g, q) = r(g, q), ∀q∈W. (3.1)

For this elliptic problem, the following regularity estimate holds (see,e.g., [3,15])

Sgγ+T g1+γ ≤Cg0, (3.2)

where γ∈(0,1] depends on the maximum interior angle ω <2πofΩ (for example,γ= ^π_ω−and γ= 1 ifΩ is convex for the Dirichlet boundary condition (1.2) case).

Then we define the corresponding discrete pair of operatorsT_handS_h T_h:W −→W_h, S_h:W −→V_h such that for anyg∈W

a(S_hg,v_h)−b(v_h, T_hg) = 0, ∀v_h∈V_h,

b(S_hg, q_h) +d(T_hg, q_h) =r(g, q_h), ∀q_h∈W_h. (3.3)

Based on the Babuˇska-Brezzi condition of the mixed finite element spaceV_h×W_h, we have

S_hgV+T_hg0≤Cg0. (3.4)

We also need to define the mixed finite element projection operator (R_h, G_h) by R_h×G_h:W×V−→W_h×V_h

such that for (p,u)∈W×V

a(G_h(u, p),v_h)−b(v_h, R_h(u, p)) = a(u,v_h)−b(v_h, p), ∀v_h∈V_h,

b(G_h(u, p), q_h) +d(R_h(u, p), q_h) =b(u, q_h) +d(p, q_h), ∀q_h∈W_h. (3.5) Notice that operator G_h and R_h are mutually coupled through system (3.5). For this type of projection, we have the following error estimate (see,e.g., [8])

G_h(u, p)−u_V+R_h(u, p)−p₀ ≤Ch^s, (3.6) where (u, p) satisfies problem (2.1) and the constantCdepends on the eigenfunction (u, p) but independent of the mesh sizeh.

So the eigenvalue problem (2.1) can be written as

λT p=p, (3.7)

and the discrete eigenvalue problem (2.8) can also be written as

λ_hT_hp_h =p_h. (3.8)

Based on the operator definitions (3.1), (3.3), and (3.5), the following relation holds

R_h(S, T) =T_h, (3.9)

where the composite operatorR_h(S, T) is defined by

R_h(S, T)g=R_h(Sg, T g), ∀g∈W,

and this type of definition is also valid for G_h(S, T). From the projection definition (3.5) and equations (3.7) and (3.8), we also have the following relations

λ_hR_h(S, T)p_h=p_h, (3.10)

and

λR_h(S, T)p=R_h(u, p). (3.11)

Throughout this paper, we assume the following hypothesis holds

Sg−I_hSg0→0, ash→0 for anyg∈L²(Ω). (3.12) Based on the abstract theory of [2,20], the recent important results [4,5], and [6], Lemma 6.1, and (2.6), (2.7) and (3.12), we know a priori error estimates for the eigenpair approximation (u_h, p_h, λ_h) of (2.8) whenhis small

enough,

|λ−λ_h| ≤Ch^2s,

u−u_hV+p−p_h0≤Ch^s, (3.13)

wheres= min{k, t}.

For the aim of the superconvergence analysis for the eigenfunction projectionR_h(u, p) and the eigenfunction approximationp_h, we need the following superconvergence betweenJ_hpandR_h(u, p) which has been analyzed by Douglas and Roberts in 1985 (see [10]) and Brezzi and Fortin in 1991 (see [8]).

Lemma 3.1 (cf.[8]).Assume the mixed finite element spaceV_h×W_hsatisfy the commuting diagram(2.5). We have the following superconvergence result for the mixed finite element projectionR_h(u, p)and the corresponding interpolant J_hp

R_h(u, p)−J_hp0 ≤Ch^s+γ. (3.14)

Proof. First we choose a functionφsuch thatφ_r= 1 andR_h(u, p)−J_hp_r=r(R_h(u, p)−J_hp, φ). Then we define the auxiliary equation:

Find (z, ψ)∈V×W such that

a(v,z)−b(v, ψ) = 0, ∀v∈V,

b(z, q) +d(q, ψ) =r(q, φ), ∀q∈W. (3.15)

This auxiliary equation has the following regularity ([3,15])

zγ+ψ1+γ ≤Cφr. (3.16)

The corresponding mixed finite element approximation (z_h, ψ_h)∈V_h×W_h is defined as:

a(v_h,z_h)−b(v_h, ψ_h) = 0, ∀v_h∈V_h,

b(z_h, q_h) +d(q_h, ψ_h) =r(q_h, φ), ∀q_h∈W_h. (3.17) The approximation (z_h, ψ_h) has the following error estimate

z−z_h0+ψ−ψ_h0≤Ch^γφ_r. (3.18)

Then with commuting diagram property (2.5) and equations (3.5), (3.15), (3.17), we have R_h(u, p)−J_hp_r=r(R_h(u, p)−J_hp, φ)

=b(z_h, R_h(u, p)−J_hp) +d(R_h(u, p)−J_hp, ψ_h)

=b(z_h, R_h(u, p)−p) +d(R_h(u, p)−J_hp, ψ_h)

=a(G_h(u, p)−u,z_h) +d(R_h(u, p)−J_hp, ψ_h)

=a(G_h(u, p)−u,z_h−z) +a(G_h(u, p)−u,z) +d(R_h(u, p)−J_hp, ψ_h)

=a(G_h(u, p)−u,z_h−z) +b(G_h(u, p)−u, ψ) +d(R_h(u, p)−J_hp, ψ_h)

=a(G_h(u, p)−u,z_h−z) +b(G_h(u, p)−u, ψ−ψ_h) +b(G_h(u, p)−u, ψ_h) +d(R_h(u, p)−p, ψ_h) +d(p−J_hp, ψ_h)

=a(G_h(u, p)−u,z_h−z) +b(G_h(u, p)−u, ψ−ψ_h) +d(p−J_hp, ψ_h). (3.19) From (3.6) and (3.18), it is easy to know that

|a(G_h(u, p)−u,z_h−z)| ≤Ch^s+γ, (3.20)

|b(G_h(u, p)−u, ψ−ψ_h)| ≤Ch^s+γ. (3.21) We can also estimate the termd(p−J_hp, ψ_h) as

|d(p−J_hp, ψ_h)| ≤

Ωϕ(p¯ −J_hp)ψ_hdΩ +

Ω(ϕ−ϕ)(p¯ −J_hp)ψ_hdΩ

≤Ch^s+1ϕ1, (3.22)

where ¯ϕis the interplant ofϕonto the piecewise constant function. Combining (3.19), (3.20), (3.21), and (3.22), we can obtain

Rh(u, p)−J_hpr≤Ch^s+γ.

Combining the above inequality with the equivalence of norms · 0 and · r, the desired result (3.14) can be

derived.

Now, let us state the superconvergence result which is the main content in this section. The idea we use here comes from [18] which is based on the operator analysis.

Theorem 3.2. Let (u_h, p_h, λ_h) be the corresponding discrete eigenpair approximation for the eigenvalue prob- lem (2.8)and (G_h(u, p), R_h(u, p)) be the mixed finite element projection of (u, p)defined by (3.5)where (u, p) is the solution of(2.1). When the mesh sizehis small enough, we have the following superconvergence result

R_h(u, p)−p_h0 ≤Ch^s+γ, (3.23)

where the constant C depends on the eigenvalueλbut independent of the mesh size h.

Proof. First from (3.7), (3.10), and (3.11), we have

(I−λT)(R_h(u, p)−p_h) = (λ_hR_h(S, T)−λT)(R_h(u, p)−p_h) +R_h(u, p)−p_h

−λ_hR_h(S, T)(R_h(u, p)−p)−λ_hR_h(S, T)(p−p_h)

= (λ_hR_h(S, T)−λT)(R_h(u, p)−p_h) + (λ−λ_h)R_h(S, T)p

−λ_hR_h(S, T)(R_h(u, p)−p). (3.24)

Let us define a functionδ_h as

δ_h =R_h(u, p)−p_h−r(R_h(u, p)−p_h, p)p. (3.25) Since r(p, p) = 1, it is easy to check that r(R_h(u, p)−p_h−r(R_h(u, p)−p_h, p)p, p) = 0. Hence the following inequality holds

cδhr≤ (I−λT)δ_hr. (3.26)

From (3.13), (3.4), (3.2), (3.6), (3.24), (3.26), and (I−λT)p= 0, we obtain

cδhr≤ |λh−λ|Rh(S, T)(R_h(u, p)−p_h)r+λ(Rh(S, T)−T)(R_h(u, p)−p_h)r

+|λ_h−λ|R_h(S, T)pr+λ_hR_h(S, T)(R_h(u, p)−p)r

≤Ch^2sTh(R_h(u, p)−p_h)r+Cλh^γT(R_h(u, p)−p_h)1+γ

+Ch^2sThpr+λ_hRh(S, T)(R_h(u, p)−p)r

≤Ch^2sRh(u, p)−p_hr+Cλh^γRh(u, p)−p_hr

+Ch^2sp_r+λ_hR_h(S, T)(R_h(u, p)−p)_r

≤Ch^2sp_r+Cλh^γR_h(u, p)−p_hr+λ_hR_h(S, T)(R_h(u, p)−p)_r, (3.27) where we used the error estimate

(Rh(S, T)−T)(R_h(u, p)−p_h)r≤Ch^γT(R_h(u, p)−p_h)1+γ ≤Ch^γRh(u, p)−p_hr, which is based on the regularity property (3.2).

Now let us estimate the term R_h(S, T)(R_h(u, p)−p)_r. First we choose a function g_h ∈ W_h such that g_hr = 1 and −r(g_h, R_h(S, T)(R_h(u, p)−p)) = R_h(S, T)(R_h(u, p)−p)_r. Then we define the auxiliary equation: Find (x_h, ξ_h)∈V_h×W_h such that

a(v_h,x_h)−b(v_h, ξ_h) = 0, ∀v_h∈V_h

b(x_h, q_h) +d(q_h, ξ_h) =r(g_h, q_h), ∀q_h∈W_h. (3.28)

Based on the Babuˇska-Brezzi condition of the mixed finite element spaceV_h×W_hand the equivalence between the norms · _rand · ₀, we have

x_hV+ξh0≤C sup

0=(vh,qh)∈Vh×Wh

a(v_h,x_h)−b(v_h, ξ_h) +b(x_h, q_h) +d(q_h, ξ_h) v_hV+q_h0

=C sup

0=(vh,qh)∈Vh×Wh

r(g_h, q_h) v_hV+qh0

≤Cg_hr

=C. (3.29)

Choosing (v_h, q_h) = (G_h(S, T)(R_h(u, p)−p), R_h(S, T)(R_h(u, p)−p)) in (3.28) and combining (3.1), (3.3), and (3.5) leads to

−r(gh, R_h(S, T)(R_h(u, p)−p)) =a(G_h(S, T)(R_h(u, p)−p),x_h)−b(x_h, R_h(S, T)(R_h(u, p)−p))

−b(G_h(S, T)(R_h(u, p)−p), ξ_h)−d(R_h(S, T)(R_h(u, p)−p), ξ_h)

=a(S(R_h(u, p)−p),x_h)−b(x_h, T(R_h(u, p)−p))

−b(S(R_h(u, p)−p), ξ_h)−d(T(R_h(u, p)−p), ξ_h)

=−r(R_h(u, p)−p, ξ_h). (3.30)

Furthermore from the defintion ofJ_h, the superconvergence (3.14), and inequality (3.29), we have the following estimate

|r(R_h(u, p)−p, ξ_h)|=|r(R_h(u, p)−J_hp, ξ_h) +r(J_hp−p, ξ_h)|

≤

Ωρ(J_hp−p)ξ_hdΩ

+Ch^s+γξ_h0

=

Ω

¯

ρ(J_hp−p)ξ_hdΩ +

Ω

(ρ−ρ)(J¯ _hp−p)ξ_hdΩ

+Ch^s+γξ_h0

≤Chρ1,∞Jhp−p0ξh0+Ch^s+γξh0

≤Ch^s+γ(1 +ρ1,∞)ξh0

≤Ch^s+γ(1 +ρ1,∞)ghr, (3.31)

where ¯ρdenotes the interplant of ρ onto the piecewise constant function. Then combining (3.30), (3.31), and the definition ofg_h, we obtain

R_h(S, T)(R_h(u, p)−p)_r≤Ch^s+γ(1 +ρ_1,∞). (3.32) So from (3.25), (3.27), and (3.32) and the property of functionρ, we can give the estimate forR_h(u, p)−p_h0 as

Rh(u, p)−p_hr≤Cδhr+r(Rh(u, p)−p_h, p)pr

≤Ch^s+γ+Cλh^γRh(u, p)−p_hr+C|(ρp, Rh(u, p)−p_h)|

=Ch^s+γ+Cλh^γRh(u, p)−p_hr+C|(ρp, Rh(u, p)−p)|

+C|(ρp, p−p_h)|. (3.33)

Now, let us consider the two terms|(ρp, R_h(u, p)−p)|and|(ρp, p−p_h)|. First from (2.1), (2.8), and (3.13), the following holds

|(ρp, p−p_h)|=|(ρp, p)−(ρp, p_h)|

= 1

2(ρp, p)−(ρp, p_h) +1

2(ρp_h, p_h)

= 1

2|(ρ(p−p_h), p−p_h)|

≤Ch^2s. (3.34)

Then from (2.4), (3.5), and (3.6), we have

λ(ρp, R_h(u, p)−p) =−a(u, G_h(u, p)−u) +b(G_h(u, p)−u, p) +b(u, R_h(u, p)−p) +d(p, R_h(u, p)−p)

=−a(u−G_h(u, p), G_h(u, p)−u) +b(G_h(u, p)−u, p−R_h(u, p)) +b(u−G_h(u, p), R_h(u, p)−p) +d(p−R_h(u, p), R_h(u, p)−p)

≤Ch^2s. (3.35)

Finally combining (3.33), (3.34), and (3.35) leads to

R_h(u, p)−p_hr ≤Ch^s+γ+Cλh^γR_h(u, p)−p_hr. (3.36) It means whenhis small enough, we have

R_h(u, p)−p_hr ≤Ch^s+γ. (3.37)

The desired result (3.23) can be obtained by the equivalence between the norms · r and · 0, and the proof

is complete.

Based on the obtained superconvergence (3.14) and (3.23), we give some obvious corollaries.

Corollary 3.3. For the eigenfunction approximation p_h of the eigenvalue problem (2.4) by the mixed finite element method and the L²-projection J_hp of the exact eigenfunction, the following superconvergence result holds when the mesh size his small enough

p_h−J_hp₀ ≤Ch^s+γ. (3.38)

Based on this superconvergence result, using a suitable interpolation postprocessing operator forp_h, we can get a superconvergent eigenfunction approximation.

Corollary 3.4. Let(u_h, p_h, λ_h)be the corresponding discrete eigenpair approximation for the eigenvalue prob- lem (2.8)and (G_h(u, p), R_h(u, p))is the mixed finite element projection of (u, p) defined by(3.5) where (u, p) is the exact solution of (2.1). When the mesh size h is small enough, we have the following superconvergence result

u_h−G_h(u, p)V+R_h(u, p)−p_h0 ≤Ch^s+γ, (3.39) where constantC depends on the eigenvalueλbut independent of the mesh size h.

Proof. From (2.4), (2.8), (3.5), (3.38) and Babuˇska-Brezzi condition, we have u_h−G_h(u, p)_V+p_h−R_h(u, p)₀

≤C sup

0=(vh,qh)∈Vh×Wh

a(u_h−G_h(u, p),v_h)−b(v_h, p_h−R_h(u, p)) +b(u_h−G_h(u, p), q_h) +d(p_h−R_h(u, p), q_h) v_hV+q_h0

=C sup

0=(vh,qh)∈Vh×Wh

r(λ_hp_h−λp, q_h) v_hV+q_h0

=C sup

0=(vh,qh)∈Vh×Wh

λ_hr(p_h−J_hp, q_h) +λ_hr(J_hp−p, q_h) + (λ_h−λ)r(p, q_h) v_hV+qh0

≤C(ph−J_hp0+hJhp−p+|λ−λ_h|)

≤Ch^s+γ, (3.40)

where we use the same technique as in (3.31).

Remark 3.5. From (3.39), we know that there exists supercovergence for both G_h(u, p) − u_hV and Rh(u, p)−p_h0 for general second order elliptic eigenvalue problems by general mixed finite element methods.

This is an extension of the results in [11–13].

4. Numerical results

In this section, we give three numerical examples to illustrate the superconvergence between the mixed finite element approximation (u_h, p_h) and the corresponding mixed finite element projection (G_h(u, p), R_h(u, p)).

Furthermore the superconvergence between p_h and interpolantJ_hpis also investigated here. Throughout this section, numerical results are given for the first eigenvalue and its corresponding eigenfunction. Of course, we should point out that our result is also valid for other eigenvalues. Here, the implicitly restarted Arnoldi method is applied to compute the eigenvalues. The Raviart-Thomas, Brezzi-Douglas-Marini and Brezzi-Douglas-Fortin- Marini mixed finite elements are applied to solve the eigenvalue problems on the triangular and rectangular meshes. The notation used in the error figures is defined as follows

R_hp:=R_h(u, p),andG_hu=G_h(u, p), where (u, p) is the exact solution of the corresponding eigenvalue problem.

4.1. Model eigenvalue problem

In this subsection, we investigate the superconvergence phenomena of the model eigenvalue problem

⎧⎨

⎩

−Δp=λp, in Ω, p= 0, on∂Ω,

Ωp²dΩ= 1,

(4.1)

on the domainΩ= (0,1)×(0,1).

It is well known that the exact eigenpair is

p_kl= sin(kπx) sin(lπy), λ_kl= (k²+l²)π², 0≤k, l andk, l are integers.

Here we test the superconvergence property on triangular and rectangular meshes. On the triangular meshes, the lowest and first order Raviart-Thomas (RT₀andRT₁) and the lowest order Brezzi-Douglas-Marini (BDM₁) mixed finite elements are applied to do the test. Additionally we also use the lowest and first order Raviart- Thomas (RT₀ and RT₁), the first order Brezzi-Douglas-Fortin-Marini (BDF M₁) mixed finite elements on the rectangular meshes to solve the eigenvalue problem (4.1).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1. Initial triangular and rectangular meshes for regular refinement.

10⁰ 10¹ 10² 10³

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

0

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||p_h−R_hp||₀

||uh−G

hu||

V

slope=−1 slope=−2

10⁰ 10¹ 10² 10³

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by BDM

1

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||ph−R

hp||

0

||uh−G

hu||

V

slope=−1 slope=−2

10⁰ 10¹ 10²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

1

|λ−λ_h|

||ph−p||

0

||ph−J hp||

0

||ph−R hp||

0

||uh−G hu||

V slope=−2 slope=−3 slope=−4

Figure 2. Errors of RT₀, BDM₁ and RT₁ on the regular refinement triangular meshes for model problem.

First we give the numerical results of the superconvergence on the triangular meshes obtained by regular refinement of the initial mesh showed in Figure1. The corresponding numerical results withRT₀, BDM₁ and RT₁are presented in Figure2. We also test theRT₀,RT₁andBDF M₁mixed finite elements on the rectangular meshes. Right hand side of Figure1shows the initial mesh and we use regular refinement to produce the mesh sequence. The numerical results are showed in Figure3.

In order to test the superconvergence on more general meshes. We also test the superconvergence on the mesh sequence in which each level triangular mesh is obtained by Delaunay algorithm. The triangular mesh sequence is presented in Figure4. The corresponding numerical results are listed in Figure 5.

10⁰ 10¹ 10² 10³ 10⁻⁶

10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

0 on rectangular mesh

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||p_h−R_hp||₀

||uh−G

hu||

V

slope=−1 slope=−2

10⁰ 10¹ 10²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

1 on rectangular mesh

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||ph−R

hp||

0

||uh−G

hu||

V

slope=−2 slope=−4

10⁰ 10¹ 10²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by BDFM

1 on rectangular mesh

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||p_h−R_hp||₀

||uh−G

hu||

V

slope=−2 slope=−4

Figure 3.Errors ofRT₀,RT₁ andBDF M₁on the regular refinement rectangular meshes for model problem.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4. Unstructured triangular mesh sequence.

10⁰ 10¹ 10² 10⁻⁶

10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

0 on irregular refinement mesh

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||ph−R

hp||

0

||uh−G

hu||

V

slope=−1 slope=−2

10⁰ 10¹ 10²

10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by BDM

1 on irregular refinement mesh

|λ−λ_h|

||ph−p||

0

||ph−J

hp||

0

||ph−R

hp||

0

||uh−G

hu||

V

slope=−1 slope=−2

10⁰ 10¹ 10²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

log(1/h)

Errors

Errors by RT

1 on irregular refinement mesh

|λ−λ_h|

||p_h−p||₀

||p_h−J_hp||₀

||p_h−R_hp||₀

||u_h−G_hu||_V slope=−2 slope=−3 slope=−4

Figure 5. Errors ofRT₀, BDM₁ andRT₁ on the irregular refinement triangular meshes for model problem.

From Figures2,3, and5, we can find that there exists superconvergence for the eigenfunction approximation (u_h, p_h) and the corresponding projection (G_h(u, p), R_h(u, p)), the approximationp_hand interpolantJ_hp. This confirms the results of Corollaries3.3and3.4.

4.2. More general eigenvalue problem

In this subsection, we are concerned with a more general eigenvalue problem (1.1) with Dirichlet boundary condition (1.2), and

A=

1 +

x₁−¹₂₂

x₁−¹₂ x₂−¹₂ x₁−¹₂ x₂−¹₂

1 +

x₂−¹₂₂

, ϕ=e^{(x1−12)(x2−12)}, ρ= 1 + (x₁−¹₂)(x₂−¹₂) andΩ= (0,1)×(0,1).

Since the exact solution is not known, we choose an adequately accurate approximationλ= 23.7784248452082 which is obtained by extrapolation method (see, e.g., [17,19]) as the exact first eigenvalue (has 8 significant digits at least). In order to measure the errors of eigenfunction approximations, we use the corresponding higher order Galerkin finite element eigenfunction approximation, which has higher order accuracy in L²(Ω) norm sense, as the exact eigenfunction.

First we test the superconvergence on triangular meshes obtained by the regular refinement with the initial mesh showed in Figure1. The corresponding numerical results byRT₀,BDM₁, andRT₁ mixed finite elements are presented in Figure6. Then we give the numerical results ofRT₀,RT₁andBDF M₁ mixed finite elements on rectangular meshes which are also obtained by regular refinement with the initial mesh showed in Figure1.

Figure7 shows the corresponding numerical results byRT₀,RT₁, andBDF M₁.

From Figures6and7, we can find there also exists superconvergence of the general problem and this confirms the results of Corollaries3.3and3.4 again.

Remark 4.1. From Figures 2, 3, 5, 6, and 7, it seems that there exists the ultrasuperconvergence (ph− R_h(u, p)0 ≤Ch^s+2) for p_h−R_h(u, p) by high order mixed finite element method. The reason may rest with γ= 2 for high order mixed finite element methods on the unit square domain.