A new nonconforming mixed finite element scheme for the second-order elliptic problem is proposed based on a new mixed variational form. It has the lowest degrees of freedom on rectangular meshes. The superclose property is proven by employing integral identity technique. Then global superconvergence result is derived through interpolation postprocessing operators. At last, some numerical experiments are carried out to verify the theoretical analysis.
1. Introduction
Mixed finite element method (MFEM) is an important branch of FEMs and has been used widely in numerical computation of practical problems. A lot of studies on this aspect have been devoted to the second-order elliptic problems [1–5], in which two approximation spaces of MFEM should satisfy the famous B-B condition [6]. However, since the variable u and flux ψ=∇u belong to L2(Ω) and H(div,Ω), respectively, it is not easy to construct a stable MFE space pair. In order to circumvent or ameliorate this deficiency, many approaches have been proposed, such as the least squares FEM [7], stabilization FEM [8], and H1-Galerkin FEM [9]. Recently, [10] presented a new MFEM, in which B-B condition is automatically satisfied when Mh and Hh meet a relation of inclusion; that is, ∇Mh⊂Hh, where Mh and Hh are finite element approximation spaces of u and flux ψ, respectively. This advantage makes the construction of stable MFE space pair extremely simple and convenient. A family of triangular and rectangular conforming MFE space pairs with lower degrees of freedom is constructed in [10], in which the total degrees of freedom of first-order and second-order MFE schemes are about 5Np and 16Np, respectively; herein Np denotes the number of nodal points in subdivision. Later, [11] derived the similar results with [10] for conforming MFEM and gave a numerical example.
In this paper, motivated by the idea of [10, 11], we first construct a new nonconforming MFEM (NMFEM). The original variable u is approximated by the constrained Q1rot element space [12] and the flux ψ by piecewise constant vectors space, respectively. Note that the total degrees of freedom of the NMFEM are only about 3Np, the lowest on rectangular meshes. We prove that it satisfies B-B condition. Then by the use of integral identity technique, we derive the superclose property for u in energy norm and flux ψ in L2 norm. Furthermore, the global superconvergence result with order O(h2) is obtained through interpolation postprocessing operators. Finally, some numerical results are provided to verify the theoretical analysis. It is observed that, compared with FEM using 4-node quadrilateral (FEM-Q4) and MFEM (u and flux ψ are approximated by piecewise constants and the Raviart-Thomas element, resp.), NMFEM behaves well and has higher rates than MFEM, and it has almost the same rate as FEM-Q4 for u in L2 norm, but a higher rate than FEM-Q4 for u in energy norm. Moreover, NMFEM is effective and accurate for the diffusion problem studied in [13].
The remainder of this paper is organized as follows. In Section 2, we introduce NMFEM and derive the superclose property and superconvergence results. In Section 3, we carry out some numerical experiments to show the performance of NMFEM.
We will use standard notations for the Sobolev spaces Hm(Ω) with norm ∥·∥m and seminorm |·|m, Hm(K) with norm ∥·∥m,K and seminorm |·|m,K, where m>0 is an integer. Besides, let ∥·∥0 and ∥·∥0,K be the L2(Ω) norm and L2(K) norm, respectively. Throughout the paper, C denotes a positive constant independent of the mesh parameter h and may be different at each appearance.
2. Superconvergence Analysis for NMFEM
Consider the following elliptic problem:
(1)-Δu=f,inΩ,u=0,on∂Ω,
where Ω⊂R2 is a bounded convex polygon domain, f∈L2(Ω).
Let ψ=∇u, and then problem (1) is equivalent to the following equations:
(2)ψ-∇u=0,inΩ,-divψ=f,inΩ,u=0,on∂Ω.
We adopt a new mixed variational form in [10] of problem (2). Find (ψ,u)∈H×M such that
(3)a(ψ,φ)+b(φ,u)=0,∀φ∈H,b(ψ,v)=G(v),∀v∈M,
where H=(L2(Ω))2, M=H01(Ω), a(ψ,φ)=∫Ωψ·φdxdy, b(φ,v)=-∫Ωφ·∇vdxdy, G(v)=-∫Ωfvdxdy.
Obviously, a(·,·) and b(·,·) are continuous bilinear functionals, G(·) is a continuous linear functional, and for all φ∈H,a(φ,φ)=∫Ωφ·φdxdy=∥φ∥02. Moreover, for v∈M, we have ∇v∈H. So there exists a constant β>0 such that
(4)supφ∈Hb(φ,v)∥φ∥0≥b(-∇v,v)∥-∇v∥0=|v|1≥β∥v∥1,
that is, the B-B condition is satisfied, and therefore (3) has a unique solution (ψ,u).
Let Q((ψ,u),(φ,v))=a(ψ,φ)+b(φ,u)-b(ψ,v), and then (3) can be written as
(5)Q((ψ,u),(φ,v))=-G(v),∀φ∈H,v∈M.
Let Th be a rectangular partition of the domain Ω. For a given element K∈Th, its four vertices are denoted by a1=(xK-hx,yK-hy), a2=(xK+hx,yK-hy), a3=(xK+hx,yK+hy), a4=(xK-hx,yK+hy), and four edges by li=aiai+1¯(i=1,2,3,4mod4), hK=maxK∈Th{hx,hy},h=maxK∈ThhK. Let K^=[-1,1]×[-1,1] be the reference element with nodes a^1=(-1,-1), a^2=(1,-1), a^3=(1,1), a^4=(-1,1) and edges l^i=a^ia^i+1¯(i=1,2,3,4mod4).
Define the affine mapping F:K^→K by
(6)x=xK+hxξ,y=xK+hyη.
Let P1(K^) be the space of polynomials with degrees ≤1 defined on K^, and then the constrained Q1rot element space CR0h is defined by [12]:
(7)CR0h={∫lv;v|K=v^∘F-1,v^∈P1(K^),∫l[v]ds=0,l⊂∂K,∀K∈Th},
where [v] denotes the jump value of v across the boundary l, and [v]=v if l⊂∂Ω.
Let NiV denote the number of interior nodes. It has been proven in [12] that dim(CR0h)=NiV and P1(K^)=span{ϕ^i,i=1~4}, where ϕ^i are defined associated with nodes a^i(i=1,2,3,4) of K^ as
(8)ϕ^1=14(1-ξ-η),ϕ^2=14(1+ξ-η),ϕ^3=14(1+ξ+η),ϕ^4=14(1-ξ+η).
We choose the following FE spaces Mh and Hh to approximate M and H, respectively:
(9)Mh=CR0h,Hh={q=(q1,q2);q|K∈(Q0(K))2,∀K∈Th},
where Q0(K) is the space of constants on K. Obviously, the total degree of freedoms of the nonconforming MFE space pair Hh×Mh is 3Np, and ∥·∥h=(∑K∈Th|·|1,K2)1/2 is a norm over Mh.
Then the MFE approximation of problem (3) is to find (ψh,uh)∈Hh×Mh such that
(10)a(ψh,φh)+bh(φh,uh)=0,∀φh∈Hh,bh(ψh,vh)=G(vh),∀vh∈Mh,
where bh(φh,vh)=-∑K∈Th∫Kφh·∇vhdxdy.
For vh∈Mh, from the affine mapping F and the definition of Mh, we can get vh|K∈P1(K), and then ∇vh|K∈(Q0(K))2, that is, ∇vh∈Hh. Thus
(11)supφh∈Hhbh(φh,vh)∥φh∥0≥bh(-∇vh,vh)∥-∇vh∥0=∑K∈Th∫K∇vh·∇vhdxdy∥-∇vh∥0=∥vh∥h2∥vh∥h=∥vh∥h,
that is, the discrete B-B condition holds, and (10) has a unique solution (ψh,uh).
For all (qh,λh)∈Hh×Mh, let Qh((qh,λh),(φh,vh))=a(qh,φh)+bh(φh,λh)-bh(qh,vh), and then we can prove the following two important lemmas.
Lemma 1.
For all (qh,λh)∈Hh×Mh, the following inequality holds:
(12)sup(φh,vh)∈Hh×MhQh((qh,λh),(φh,vh))∥φh∥0+∥vh∥h≥C(∥qh∥0+∥λh∥h).
Proof.
In order to prove (12), we will construct a pair (φh,vh)∈Hh×Mh satisfying
(13)Qh((qh,λh),(φh,vh))≥C(∥qh∥0+∥λh∥h)(∥φh∥0+∥vh∥h).
Obviously,
(14)Qh((qh,λh),(qh,λh))=∥qh∥02.
On the other hand, for a given arbitrary but fixed λh∈Mh, by the discrete B-B condition, there exists a wh∈Hh such that
(15)∑K∈Th∫Kwh·∇λhdxdy≥∥wh∥0∥λh∥h,∥wh∥0=∥λh∥h.
So, for 0<r<1, there holds
(16)Qh((qh,λh),(-rwh,0))=-ra(qh,wh)+r∑K∈Th∫Kwh·∇λhdxdy≥-r∥qh∥0∥λh∥h+r∥λh∥h2≥-r∥qh∥02+34r∥λh∥h2.
As a result, setting (φh,vh)=(qh-rwh,λh), we have
(17)Qh((qh,λh),(qh-rwh,λh))=Qh((qh,λh),(qh,λh))+Qh((qh,λh),(-rwh,0))≥(1-r)∥qh∥02+34r∥λh∥h2,
which implies that
(18)Qh((qh,λh),(qh-rwh,λh))≥C1(∥qh∥02+∥λh∥h2)≥C12(∥qh∥0+∥λh∥h)2,
where C1=min{1-r,(3/4)r}.
Note that
(19)∥qh-rwh∥0+∥λh∥h≤∥qh∥0+r∥wh∥0+∥λh∥h=∥qh∥0+(1+r)∥λh∥h≤(1+r)(∥qh∥0+∥λh∥h),
and we have
(20)Qh((qh,λh),(qh-rwh,λh))≥C12(1+r)(∥qh∥0+∥λh∥h)(∥qh-rwh∥0+∥λh∥h),
which follows the desired result (12).
Let Πh and Ih denote the associated interpolation operators of Mh and conforming bilinear element space, respectively. Let πh be the Q1rot element interpolation operator [14]; that is, for all v∈H1(K), πhv∈Q1rot satisfying ∫lπhvds=∫lvds, where Q1rot={vh;vh|K∈span{1,x,y,x2-y2},∫l[vh]ds=0,l⊂∂K}. It has been shown in [12] that, for all v∈H2(K), Πhv=πhIhv∈Mh because xy is a bubble function for πh.
Lemma 2.
Assume that u∈H3(Ω), ψ∈(H2(Ω))2, we have
(21)bh(ψ-Jhψ,vh)=0,∀vh∈Mh,(22)∑K∈Th∫K∇(u-Πhu)·φhdxdy≤Ch2∥u∥3∥φh∥0,∀φh∈Hh,(23)∑K∈Th∫∂Kψ·nvhds≤Ch2|ψ|2∥vh∥h,∀vh∈Mh,
where Jh:(L2(Ω))2→Hh is a interpolation operator satisfying ∫K(ψ-Jhψ)dxdy=0.
Proof.
Since (23) has been proven by one of the authors in [15], we only need to prove (21) and (22).
In fact, because ∂vh/∂x and ∂vh/∂y are constants on each K∈Th, we have
(24)bh(ψ-Jhψ,vh)=-∑K∈Th∫K(ψ-Jhψ)·∇vhdxdy=0,
which is (21).
On the other hand, note that
(25)∫K∇(u-Πhu)·φhdxdy=∫K∇(u-Ihu)·φhdxdy+∫K∇(Ihu-Πhu)·φhdxdy.
Since φh=(φ1h,φ2h) is a constant vector on K and Πhv=πhIhv, it follows from integration by parts that
(26)∫K∇(Ihu-Πhu)·φhdxdy=φh|K∫∂K(Ihu-πhIhu)ds=φh|K∫∂K(Ihu-Ihu)ds=0.
Furthermore, let F(y)=(1/2)((y-yK)2-hy2). Note that (u-Ihu)(xK±hx,yK±hy)=0 and F(yK±hy)=0. Employing integral identity technique [16], we have
(27)∫K∂(u-Ihu)∂xφ1hdxdy=φ1h|K∫KF′′(y)∂(u-Ihu)∂xdxdy=φ1h|K(F′(yK+hy)∫l3∂(u-Ihu)∂xdx-F′(yK-hy)∫l1∂(u-Ihu)∂xdx-∫KF′(y)∂2(u-Ihu)∂x∂ydxdy)=-φ1h|K(F(yK+hy)∫l3∂2(u-Ihu)∂x∂ydx-F(yK-hy)∫l1∂2(u-Ihu)∂x∂ydx-∫KF(y)∂3(u-Ihu)∂x∂y2dxdy)=φ1h|K∫KF(y)∂3(u-Ihu)∂x∂y2dxdy≤Ch2∥u∥3,K∥φ1h∥0,K.
Similarly,
(28)∫K∂(u-Ihu)∂yφ2hdxdy≤Ch2∥u∥3,K∥φ2h∥0,K.
Therefore, from (25)–(28), we have
(29)∑K∈Th∫K∇(u-Πhu)·φhdxdy=∑K∈Th∫K∇(u-Ihu)·φhdxdy=∑K∈Th(∫K∂(u-Ihu)∂xφ1hdxdy+∫K∂(u-Ihu)∂yφ2hdxdy)≤Ch2∥u∥3∥φh∥0.
The proof is completed.
Now we start to state the following superclose property.
Theorem 3.
Assume that (ψ,u) and (ψh,uh) are the solutions of (3) and (10), respectively; u∈H3(Ω), ψ∈(H2(Ω))2, there holds
(30)∥ψh-Jhψ∥0+∥uh-Πhu∥h≤Ch2(∥u∥3+∥ψ∥2).
Proof.
For (φh,vh)∈Hh×Mh, from (2) and (10), we have
(31)a(ψ,φh)+bh(φh,u)=a(ψh,φh)+bh(φh,uh)=0,bh(ψ,vh)-bh(ψh,vh)=bh(ψ,vh)-G(vh)=-∑K∈Th∫Kψ·∇vhdxdy+∑K∈Th∫Kfvhdxdy=-∑K∈Th∫∂Kψ·nvhdxdy+∑K∈Th∫K(divψ+f)vhdxdy=-∑K∈Th∫∂Kψ·nvhdxdy.
Applying (31) yields
(32)Qh((ψ-ψh,u-uh),(φh,vh))=∑K∈Th∫∂Kψ·nvhdxdy,∀(φh,vh)∈Hh×Mh.
Using (12) in Lemma 1 and (32), we can obtain
(33)∥ψh-Jhψ∥0+∥uh-Πhu∥h≤Csup(φh,vh)∈Hh×MhQh((ψh-Jhψ,uh-Πhu),(φh,vh))∥φh∥0+∥vh∥h≤Csup(φh,vh)∈Hh×Mh((∥φh∥0+∥vh∥h)-1(Qh((ψ-Jhψ,u-Πhu),(φh,vh))-∑K∈Th∫∂Kψ·nvhdxdy)×(∥φh∥0+∥vh∥h)-1)=Csup(φh,vh)∈Hh×Mh(((∥φh∥0+∥vh∥h)-1a(ψ-Jhψ,φh)+bh(φh,u-Πhu)-bh(ψ-Jhψ,vh)-∑K∈Th∫∂Kψ·nvhdxdy)×(∥φh∥0+∥vh∥h)-1).
Hence the desired result follows from the interpolation theorem and Lemma 2.
In order to derive global superconvergence for u and flux ψ, we introduce the following postprocessing operators I2h:u∈(H2(Ω)∩H01(Ω))↦I2hu∈Q2(κ) and J2h:ψ∈(H1(Ω))2↦J2hψ∈(Q1(κ))2 as I2hu(ai)=u(ai), i=1,…,9, and ∫Kj(ψ-J2hψ)=0, j=1,2,3,4, where u(ai) are the value of u on the nodes ai and ai are nodes of Th on macroelement κ, while κ∈T2h consists of the four small elements Kj in Th(j=1,2,3,4), and Q1(κ) and Q2(κ) are bilinear and biquadratic piecewise polynomials spaces, respectively. It can be checked that the following properties hold:
(34)I2hΠhu=I2hu,∥I2hu-u∥h≤Ch2|u|3,∥I2hvh∥h≤C∥vh∥h,∀vh∈Mh,J2hJhψ=J2hψ,∥J2hψ-ψ∥0≤Ch2|ψ|2,∥J2hφh∥0≤C∥φh∥0,∀φh∈Hh.
Then we can have the following superconvergence result.
Theorem 4.
Under the assumptions in Theorem 3, there holds
(35)∥u-I2huh∥h+∥ψ-J2hψh∥0≤Ch2(∥u∥3+∥ψ∥2).
Proof.
It follows from (30) in Theorem 3, (34), and the triangle inequality that
(36)∥u-I2huh∥h+∥ψ-J2hψh∥0≤∥u-I2hΠhu∥h+∥I2hΠhu-I2huh∥h+∥ψ-J2hJhψ∥0+∥J2hJhψ-J2hψh∥0≤∥u-I2hΠhu∥h+∥I2h(Πhu-uh)∥h+∥ψ-J2hJhψ∥0+∥∥J2h(Jhψ-ψh)∥0≤∥u-I2hu∥h+C∥Πhu-uh∥h+∥ψ-J2hψ∥0+C∥Jhψ-ψh∥0≤Ch2(∥u∥3+∥ψ∥2),
which is the desired result.
3. Numerical Experiments
In this section, some numerical examples and comparison with other methods are presented to confirm theoretical analysis and good performance of NMFEM.
We consider the problem (1) with Ω=[0,1]2 and the exact solution is u=xy(1-x)(1-y), and then the flux field ψ=(ψ1,ψ2) can be expressed as ψ1=y(1-2x)(1-y),ψ2=x(1-x)(1-2y). We divide the domain Ω into a family of quasiuniform rectangles with number of N. The figures of exact solution (u,ψ) of problem (1) and finite element approximation (uh,ψh) with N=642 are plotted in Figures 1, 2, and 3, respectively.
The exact flux ψ1 (a) Numerical approximation ψ1h (b).
The exact flux ψ2 (a) and numerical approximation ψ2h (b).
The exact displacement u (a) and numerical approximation uh (b).
In Tables 1 and 2, we present the superclose and superconvergence results of the original variable u in energy norm and flux ψ in L2 norm with N=82;162;322;642;1282;2562, respectively. It is clearly that ∥ψh-Jhψ∥0, ∥uh-Πhu∥h, ∥ψ-J2hψh∥0, and ∥u-I2huh∥h are converged at order 2 with respect to h, which coincide with our theoretical analysis in Theorems 3 and 4. In order to describe the results more intuitively, we plot the errors in the logarithm scales in Figure 4.
Superclose results of ψ and u.
N
∥ψh-Jhψ∥0
Rate
∥uh-Πhu∥h
Rate
82
0.00210961611276
/
0.00313196226739
/
162
5.305340499025525e-004
1.99146324331851
7.857858400185900e-004
1.99485875617198
322
1.328284185506180e-004
1.99788150355682
1.966221092417072e-004
1.99871061529750
642
3.321927509970684e-005
1.99947134660916
4.916651922457212e-005
1.99967742778650
1282
8.305579254650728e-006
1.99986789716594
1.229231701395274e-005
1.99991934319524
2562
2.076442340906698e-006
1.99996697808428
3.073122207277826e-006
1.99997983495344
Superconvergence results of ψ and u.
N
∥ψ-J2hψh∥0
Rate
∥u-I2huh∥h
Rate
82
0.00438686071544
/
0.00401808068890
/
162
0.00109136640979
2.00705335488651
9.832140356840235e-004
2.03092911988780
322
2.724757835149567e-004
2.00193562594041
2.445206258258704e-004
2.00754934612234
642
6.809546234729317e-005
2.00049744480245
6.105100787093085e-005
2.00186914715453
1282
1.702238617992853e-005
2.00012537844145
1.525782546054283e-005
2.00046574790862
2562
4.255503868189354e-006
2.00003141881610
3.814148839722483e-006
2.00011631625361
The superclose results (a) and the superconvergence results (b).
Moreover, we compare the results of NMFEM with those of FEM using 4-node quadrilateral (FEM-Q4) and MFEM (u and flux ψ are approximated by piecewise constants and the Raviart-Thomas element, resp.).
The convergence rates of errors of u in L2 and energy norm are shown in Figures 5 and 6. The comparison of the flux ψ in L2 norm of NMFEM with MFEM is also given in Figure 7. It is observed that (a) the convergence rates of u and flux ψ in L2 norm of NMFEM are better than those of MFEM; (b) NMFEM has almost the same rate as FEM-Q4 for u in L2 norm, but a higher rate than FEM-Q4 for u in energy norm.
Convergence rates of u in L2 norm.
Convergence rates of u in energy norm.
Convergence rates of the flux ψ in L2 norm.
Furthermore, a numerical experiment is carried out to demonstrate the effectiveness and accuracy of NMFEM for the following diffusion problem:
(37)-div(D∇u)=f,inΩ,u=u∂,on∂Ω,
where f is the source term and f∈L2(Ω), u∂ is the boundary data, the permeability D is a symmetric tensor-valued function such that (a) D is piecewise Lipschitz-continuous on Ω and (b) the set of the eigenvalues of D is included in [λmin,λmax] (with λmin>0) for all (x,y)∈Ω.
As [13], we consider the problem (37) with Ω=[0,1]2,
(38)D=(1001)ifx≤0.5,D=(100000.01)ifx>0.5,
and the analytical solution
(39)u=cos(πx)sin(πy)ifx≤0.5,u=0.01cos(πx)sin(πy)ifx>0.5.
Let ψ=D∇u, we can get the superclose and superconvergence results of u in energy norm and flux ψ in L2 norm. The errors are listed in Tables 3 and 4 and plotted in the logarithm scales in Figure 8, respectively.
Superclose results of ψ and u.
N
∥ψh-Jhψ∥0
Rate
∥uh-Πhu∥h
Rate
82
0.02493802431889
/
0.03360261961598
/
162
0.00619633909611
2.00885917091448
0.00820721453088
2.03360913804615
322
0.00154620480519
2.00268467156460
0.00203799536238
2.00974189629474
642
3.863629284378830e-004
2.00070284735607
5.086068699851279e-004
2.00252791412788
1282
9.657883351384945e-005
2.00017773032582
1.270955080172888e-004
2.00063790775102
2562
2.414396265195381e-005
2.00004455935162
3.177035666820934e-005
2.00015984995534
Superconvergence results of ψ and u.
N
∥ψ-J2hψh∥0
Rate
∥u-I2huh∥h
Rate
82
0.07704230005041
/
0.06805851358971
/
162
0.01905713318621
2.01531966736879
0.01670893020781
2.02615627537278
322
0.00474917374465
2.00458266513090
0.00415565843089
2.00747038382646
642
0.00118603209709
2.00153348351974
0.00103721683378
2.00235955252584
1282
2.964169140807930e-004
2.00044337559203
2.591844848686839e-004
2.00066626183030
2562
7.409813186219603e-005
2.00011869741141
6.478819154402888e-005
2.00017656612731
The superclose results (a) and the superconvergence results (b).
Acknowledgments
The authors would like to express their sincere thanks to the anonymous referee for his many helpful suggestions, which contribute significantly to the improvement of the paper. The research is supported by the NSF of China (no. 10971203; no. 11271340), Research Fund for the Doctoral Program of Higher Education of China (no. 20094101110006), and Foundation of He’nan Educational Committee (no. 13B110144).
RaviartP.-A.ThomasJ. M.A mixed finite element method for 2nd order elliptic problemsFarhloulM.FortinM.A nonconforming mixed finite element for second-order elliptic problemsCockburnB.KanschatG.PerugiaI.SchötzauD.Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian gridsLuoZ.-D.Method of nonconforming mixed finite element for second order elliptic problemsShiD. Y.WangC. X.A new low-order non-conforming mixed finite-element scheme for second-order elliptic problemsBrezziF.FortinM.BochevP. B.GunzburgerM. D.Finite element methods of least-squares typeBochevP. B.DohrmannC. R.GunzburgerM. D.Stabilization of low-order mixed finite elements for the Stokes equationsShiD.-Y.WangH.-H.Nonconforming H1-Galerkin mixed FEM for Sobolev equations on anisotropic meshesChenS. C.ChenH. R.New mixed element schemes for a second-order elliptic problemShiF.YuJ. P.LiK. T.A new mixed finite element scheme for elliptic equationsHuJ.ManH. Y.ShiZ. C.Constrained nonconforming rotated Q1 element for Stokes flow and planar elasticityDroniouJ.le PotierC.Construction and convergence study of schemes preserving the elliptic local maximum principleRannacherR.TurekS.Simple nonconforming quadrilateral Stokes elementShiD. Y.PengY. C.ChenS. C.Error estimates for rotated Q1rot element approximation of the eigenvalue problem on anisotropic meshesLinQ.YanN. N.