The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.
1. Introduction
The elliptic perturbation problems, which are derived from the stationary formation of parabolic perturbation problems, such as the Cahn-Hilliard type equation, are very important in both theoretical research and applications. The finite element methods are always chosen to be the appropriate way to solve the numerical solutions (cf. [1–5]). Here, we consider the following two-dimensional linear stationary Cahn-Hilliard type equation as our model problem:
(1)ɛ2Δ2u-Δu=f,inΩ,u=∂u∂n=0,on∂Ω,
where Δ is the standard Laplace operator, Ω is a bounded polygonal domain in R2, ∂Ω is the boundary of Ω, and ɛ is a real parameter such that 0<ɛ≤1. Let ∂u/∂n denote the normal derivative of u along the boundary ∂Ω. Particularly, the differential equations (1) formally degenerate to Poisson equations (a plate model degenerates towards an elastic membrane problem) when ɛ tends to zero.
Semper considers its conforming finite element methods in [5]. The regularity of the solution is analyzed, quasioptimal global error estimates are presented when ɛ>h, and local analysis is also done by using techniques of Nitsche and Schatz [6] and Schatz and Wahlbin [7]. The author points out that the method behaves poorly when the perturbation parameter is much smaller than the mesh size by some numerical experiments.
On the other hand, it is well known that when fourth-order problems are discretized by a finite element method, the standard variational formulation will require the piecewise smooth functions in C1 space. However, it is very difficult to construct such functions, and even if we can do that, the element will be rather complicated. Hence a common approach to solve this problem is to use nonconforming finite elements which violate the C1-continuity requirement. In this case, two convergence criteria are generally employed: the Patch-Test [8] is used widely in engineers, but it is neither necessary nor sufficient; the Generalized Patch-Test [9] is proved to be the sufficient and necessary condition, while in practice it is often hard to be verified. To overcome the difficulty, the F-E-M criteria were proposed in [10] to make the test tractable.
Many successful nonconforming plate elements have been constructed (e.g., see [2, 3, 9–17]), but not all of them are convergent uniformly for (1) with respect to perturbation parameter ɛ. The very simple nonconforming Morley element (see [18]), which is convergent (cf. [19, 20]) and even has some superconvergent properties under uniform meshes for plate problems, see [21], however, is proved to be not uniformly convergent for (1) in [3] when ɛ→0, that is; it may diverge for second order problem like Poisson equation (see also [22]). It is considered that the main reason for this degeneracy is the fact that the finite element space is not a subspace of H1(Ω). Indeed, it is not of C0 type. A counterexample is given in [3]. For more discussions on this element, we refer to [3, 19, 22]. As an alternative, a new modified C0 element is proposed in [3], which is robust with respect to the parameter ɛ.
In [12], the convergence analysis of a nonconforming incomplete biquadratic rectangular plate element with the shape function space and the degrees of freedom P(K)=span{1,x,y,x2,xy,y2,x2y,xy2} and ΣK={vi,(∂v/∂n)(Bi)(i=1,2,3,4)}, respectively, is studied, where vi is the function value at the vertex ai of element K, (∂v/∂n)(Bi) is the unit outer normal derivative value at the middle point Bi of the edge li of K, and n=(nx,ny) is the unit outer normal vector to li. This element, similar to the famous triangular Morley element [3, 18, 19, 23], is also a non C0 element, and its convergence order O(h) was given based on the Generalized Patch-Test. In [24], a modified element is provided by replacing the degrees of freedom and the shape function of [12] with ΣK={vi,(1/|li|)∫li(∂v/∂n)ds(i=1,2,3,4)} and P(K)=span{1,x,y,x2,xy,y2,x3,y3}, respectively. Recently, [2] applied the modified Morley element of [24] to the fourth-order elliptic singular perturbation problem and proved the convergence uniformly in the perturbation parameter. However, all the studies above are based on the traditional regular triangulations.
In this paper we will present another improved element by using the same degrees of freedom of [2] and the same shape function space of [12]. Obviously, the above element is convergent for fourth-order plate bending problems according to FEM test in [10]. However, to our knowledge, there is no literature considering the convergence of this element for fourth-order singular perturbation problems. Here, we will show that this element is not uniformly convergent for fourth-order singular perturbation problems with respect to the perturbation parameter ɛ with a counterexample presented. Moreover, the convergence results are presented even under anisotropic meshes when the modified approximation formulation in [3] is employed.
The paper is organized as follows. The next section lists some preliminaries and the construction of the element. In Section 3, a counterexample is presented. In Section 4, the convergence results under the quasiuniform assumption and anisotropic meshes are provided. Numerical experiments are carried out in last section to confirm the theoretical analysis.
2. Premilinaries
Denote the inner product on L2(Ω) by (·,·), the usual Sobolev space, norm, and seminorm by Hm(Ω), ∥·∥m, and |·|m, respectively. The space H0m(Ω) is the closure in Hm(Ω) of C0∞(Ω). Equivalently, we have
(2)H01(Ω)={v∈H1(Ω);v|∂Ω=0},H02(Ω)={v∈H2(Ω);v|∂Ω=∂v∂n|∂Ω=0}.
Let Du be the gradient of u and let D2u=(∂2u/∂xi∂xj)2×2 be the 2×2 tensor of the second order partial derivatives (x=x1,y=x2). Define
(3)a(u,v)=∫ΩD2u:D2vdxdy,∀u,v∈H2(Ω),b(u,v)=∫ΩDu·Dvdxdy,∀u,v∈H1(Ω).
Then the weak form of (1) reads: find u∈H02(Ω), such that
(4)ε2a(u,v)+b(u,v)=(f,v),∀v∈H02(Ω).
By Green’s formula, it is easy to get
(5)∫ΩD2u:D2vdxdy=∫ΩΔuΔvdxdy,∀u,v∈H02(Ω).
However, it does not hold on nonconforming finite element spaces.
Without loss of generality, we assume that the edges of Ω are parallel to the x,y axis. For mesh size h, a rectangular triangulation 𝒯h of Ω is then formed by lines also parallel to x,y axis. Let K∈𝒯h be a rectangle with the central point (0,0), 2hx and 2hy the lengths of edges parallel to x axis and y axis, respectively, hK=max{hx,hy}, a1(-hx,-hy), a2(hx,-hy), a3(hx,hy), and a4(-hx,hy) the four vertices, li=aiai+1→ (i=1,2,3,4,mod(4)). Let K^ be a reference element in (ξ,η) plane with central point (0,0), four vertices a^1(-1,-1), a^2(1,-1), a^3(1,1), and a^4(-1,1), and four edges l^i=a^ia^i+1→, (i=1,2,3,4,mod(4)). Then there exists a reversible mapping FK:K^→K:
(6)x=hxξ,y=hyη.
On K we define the finite element (K,P(K),ΣK) as:
(7)P(K)=P2(K)∪{x2y,xy2},ΣK={vi,1|li|∫li∂v∂nds,(i=1,2,3,4)},
where P2(K) denotes the set of quadratic polynomials on element K. Then it is easy to check that P(K) can be uniquely determined by the degrees of freedom ΣK. The degrees of freedom are plotted in Figure 1.
The degrees-of-freedom of incomplete biquadratic element.
For every v∈H3(Ω), we define the interpolation operator Πh as Πhv|K=ΠKv, and ΠK satisfies
(8)v∈H3(K)⟶ΠKv∈P(K),
such that
(9)ΠKv(ai)=v(ai),1|li|∫li∂ΠKv∂nds=1|li|∫li∂v∂nds,(i=1,2,3,4).
It can be checked that
(10)ΠKv=v,∀v∈P2(K).
Let Vh be the associated finite element space defined by
(11)Vh={∫l[∂v∂n]v;v|K∈P(K),∀K∈𝒯h,v(a)=0,0∫l[∂v∂n]ds=0,∀nodea∈∂Ω,∀l⊂∂K},
where [∂v/∂n] is the jump value of ∂v/∂n on l⊂∂K and [∂v/∂n]=∂v/∂n if l⊂∂Ω.
Then the corresponding finite element approximation of (4) is as follows: find uh∈Vh, such that
(12)ε2ah(uh,vh)+bh(uh,vh)=(f,vh),∀vh∈Vh,
where for all uh,vh∈Vh,
(13)ah(uh,vh)=∑T∈𝒯h∫TD2uh:D2vhdxdy,bh(uh,vh)=∑T∈𝒯h∫TDuh·Dvhdxdy.
Next, we will present the modified discretization form of problem (4) in [2].
Let Πh1 be the interpolation operator of the Lagrange bilinear rectangular element corresponding to the triangulation 𝒯h. The modified finite element method of (4) reads as: find uh∈Vh, such that
(14)ε2ah(uh,vh)+bh(Πh1uh,Πh1vh)=(f,Πh1vh),∀vh∈Vh.
Note that the problem has a unique solution when ɛ>0, but when ɛ=0, the problem degenerates to
(15)bh(Πh1uh,Πh1vh)=(f,Πh1vh),∀vh∈Vh,
in this case, Πh1uh is uniquely determined, though the solution uh is not unique.
We introduce the same mesh dependent norm ∥·∥m,h and semi–norm |·|m,h on space Vh+Hm:
(16)∥·∥m,h=(∑K∈𝒯h∥·∥m,K2)1/2,|·|m,h=(∑K∈𝒯h|·|m,K2)1/2,000000000000000000000000000000000000m=0,1,2,3.
The energy norm is defined by
(17)∥|vh|∥ɛ,h=ε2ah(vh,vh)+bh(Πh1vh,Πh1vh).
3. A Counterexample
In this section, we construct a counterexample to show that the element presented in Section 2 is not convergent uniformly with respect to the perturbation parameter ɛ. That means if the element is applied to a nearly second order problem with the form of (1) when ɛ→0, the convergence rate of the method will deteriorate. In fact, like the Morley element, when it is applied to a second order equation like Poisson’s equation, the method will diverge.
As in [3], we consider the slightly modified reduced problem
(18)-Δu=f,inΩ,u=0,onΓD,∂u∂n=g,onΓN,
to simplify some calculations. Assume ∂Ω=ΓD∪ΓN, where ΓN and ΓD are disjoint subsets of ∂Ω. This problem is a second order problem with mixed boundary conditions, which can be regarded as the formal limit of the fourth-order problems
(19)ε2Δ2u-Δu=f,inΩ,u=0,onΓD,∂∂n(u-ε2Δu)=g,onΓN,Δu=0,on∂Ω.
Let 𝒯h be a triangulation of Ω, and let V~h be the finite element space of incomplete biquadratic plate element corresponding to the boundary conditions of (18) here; see Figure 2. Then the approximation problem to (18) reads as: find uh∈V~h, such that
(20)bh(uh,vh)=(f,vh)+〈g,vh〉,∀vh∈V~h,
where 〈g,vh〉=∫ΓNgvhds, ds denotes the arc length along ΓN. Moreover, the exact solution u of (18) satisfies
(21)bh(u,vh)=(f,vh)+〈g,vh〉+Eh(u,vh),∀vh∈V~h,
where
(22)Eh(u,vh)=bh(u,vh)-(f,vh)-〈g,vh〉.
Let ∥|·|∥h be the corresponding energy norm of problem (18), that is, ∥|·|∥h=bh(·,·). Then, employing Cauchy-Schwarz inequality (we refer to [3]), we have
(23)∥|u-uh|∥h≥supvh∈V~h|Eh(u,vh)|∥|vh|∥h.
In the following, we will choose a suitable exact solution u to prove the divergence of the method by virtue of (23).
The triangulation of Ω.
The domain Ω is taken as the unit square. To simplify the analytic process, the uniform triangulation with the mesh size 1/n is employed. Assume ΓD to be the intersection of ∂Ω with the coordinate axis, while ΓN to be the part of ∂Ω on x=1 and y=1. Hence, the functions in finite element space V~h are zeros at the vertices on the coordinate axis.
We assume that the exact solution of (18) is given by u=xy. Thus, g=x on y=1 and g=y on x=1. Obviously, u is harmonic, which means f=0. Therefore:
(24)Eh(u,vh)=bh(u,vh)-〈g,vh〉.
Note that u∈V~h, thus Πhu=u and here Πh is the finite element interpolation on V~h. Similar to the discussion in [3], it is easy to derive that finite element space V~h can be naturally decomposed into two spaces:
(25)V~h=V~hv+V~he,
where V~hv and V~he correspond to the vertex values and the edge values, respectively. In fact, the two spaces can be expressed as
(26)V~hv={vh∈V~h:∫e∂vh∂nds=0,∀e∈ℰh},V~he={vh∈V~h:vh(x)=0,∀x∈𝒳h},
where ℰh and 𝒳h represent the sets of the edges and vertices corresponding to the triangulation 𝒯h, respectively. Let uhv be the interpolant of u onto V~hv, then, according to its definition, we can get
(27)uhv(x)=u(x),∀x∈𝒳h,∫e∂uhv∂nds=0,∀e∈ℰh.
We begin to prove that the limit limh→0(|Eh(u,vh)|/∥|vh|∥h) is strictly positive, so we can show from (23) that the method is divergent.
In [3], the authors proved for Morley element, the decomposition of space V~h is orthogonal, while it does not hold any more for this incomplete biquadratic plate element. However, in the mesh fashion chosen before, for any element K∈𝒯h, the length of each edge is 2h=1/n. Let the center point of K be (x0,y0), then by the definition of the space V~hv, direct calculation implies the expression of uhv on element K:
(28)uhv=x0y0+12η(3ξ2-1)x0h+12ξ(3η2-1)y0h+ξηh2,
where
(29)ξ=(x-x0)h,η=(y-y0)h.
We denote the element [-1,1]×[-1,1] on (ξ,η) plane by the reference element K^. Apparently, the mapping from K^ to K is affine, hence we have
(30)∂uhv∂x=3ξηx0+12(3η2-1)y0+ηh,∂uhv∂y=3ξηy0+12(3ξ2-1)x0+ξh,
moreover, ∂u/∂x=y, ∂u/∂y=x, and then
(31)∫KDu·Duhvdxdy=∫K∂u∂x∂uhv∂x+∂u∂y∂uhv∂ydxdy=h2∫K^(hη+y0)(3ξηx0+12(3η2-1)y0+ηh)0000000+(hξ+x0)(3ξηy0+12(3ξ2-1)x0+ξh)dξdη=h2∫-11∫-11(h2η2+h2ξ2)dξdη=43h4,
thus
(32)limh→0bh(u,uhv)=limh→0∑K∈𝒯h43h4=limh→0n243h4=limh→013h2=0.
On the other hand, denoting the edges of Ω on y=1 and x=1 by ΓN1 and ΓN2, respectively, we have
(33)〈g,uhv〉=∫ΓNguhvds=∫ΓN1guhvds+∫ΓN2guhvds.
We first consider the first term, let those components of ℰh lying on ΓN1 be li, i=1,2,…,n, and let the corresponding element be Ki, the center point of Ki be (xi,y*), where xi=(2i-1)/2n=h(2i-1), y*=(2n-1)/2n=1-h. Then
(34)∫liguhvds=h∫l^i(hξ+xi)(xiy*+12η(3ξ2-1)xih00000000000000000+12ξ(3η2-1)y*h+ξηh2)ds^=h∫-11(hξ+xi)00000000×(xiy*+12η(3ξ2-1)xih+ξy*h+ξh2)dξ=2h(13y*h2+13h3+xi2y*),
thus
(35)∫ΓN1guhvds=∑i=1n2h((2i-12n)213(1-h)h2+13h3000000+(2i-12n)2(1-h))=∑i=1n2h(13(1-h)h2+13h3000000+h2(2i-1)2(1-h)13(1-h)h2)=13h2+2h3(1-h)×((2h+1)(h+1)6h3-2h+12h2+12h).
Therefore, by a limit process, we can obtain
(36)limh→0∫ΓN1guhvds=13.
Similarly, for ∫ΓN2guhvds, we have
(37)∫ΓN2guhvds=∑i=1n2h(13(1-h)h2+13h3+(2i-12n)2(1-h))=∫ΓN1guhvds,
this immediately leads to
(38)limh→0∫ΓN2guhvds=13.
Finally, since ∥|uhv|∥h2=bh(uhv,uhv)=∑K∈𝒯h∫K|Duhv|2dxdy, and
(39)∫K|Duhv|2dxdy=∫K(∂uhv∂x)2+(∂uhv∂y)2dxdy=h2∫K^(3ξηx0+12(3η2-1)y0+ηh)2+(3ξηy0+12(3ξ2-1)x0+ξh)2dξdη=245h2(x02+y02)+83h4,
we can verify that
(40)bh(uhv,uhv)=∑K∈𝒯h(245h2(x02+y02)+83h4)=245h42n∑i=1n(2i-1)2+23h2=45(1+2h2)+23h2.
From this expression we obtain that
(41)limh→0bh(uhv,uhv)=45.
This together with (32), (36), and (38) implies
(42)limh→0|Eh(u,uhv)|∥|uhv|∥h=limh→0|bh(u,uhv)-〈g,uhv〉|bh(uhv,uhv)1/2=2/34/5=53.
The divergence of the method is therefore a consequence of the basic lower bound (23).
4. Convergence Analysis in a Modified Discretization Form
In the last section, we provide a counterexample to show that the incomplete biquadratic plate element may diverge for a second order problem like Poisson equation, and hence, in the standard finite element approximation, the convergence can not be insured for problem (1). In [2], a new modified approximation form is presented for Morley element and another Morley type rectangular element. In this section, we will show that the incomplete biquadratic plate element is convergent for problem (1) uniformly with respect to the parameter ɛ under anisotropic meshes.
To begin with, we introduce the following error estimate regarding the operator Πh1 on anisotropic meshes, as to its proof, we refer to [25].
Lemma 1.
For the bilinear interpolation operator Πh1, for all u∈H2(Ω), there holds
(43)∥u-Πh1u∥0,Ω+h∥∇(u-Πh1u)∥0,Ω≤Ch2|u|2,Ω,
where C is a constant independent of triangulation.
Denote the quadratic part of the interpolant function Π^v^ by Π^v^¯ and the corresponding part of the function ΠKv on the element K by ΠKv¯, then we can get the following lemma.
Lemma 2.
For any K∈𝒯h, for all v∈H2(K), without the regular or quasiuniform assumption, we have the following estimate:
(44)|v-ΠK1ΠKv¯|1,K≤Ch|v|2,K,∀v∈H2(K).
Proof.
Apparently, ΠK1ΠKv¯ can be considered as an interpolant of v on Q1(K). For simplicity, we denote it by Π~v. We first show that on the reference element K^, for multi-index |α|=1,
(45)∥D^α(v^-Π~^v^)∥0,K^≤C|D^αv^|1,K^.
For the convenience of notations, we denote degrees-of-freedom by v(ai)=vi, (1/|li|)∫li(∂v/∂n)ds=vi+4, i=1,2,3,4. Correspondingly, v^(a^i)=v^i, (1/|l^i|)∫l^i(∂v^/∂n)ds=v^i+4, i=1,2,3,4. Hence, for all v^∈H1(K^), direct computation provides the expression of Π~^v^:
(46)Π~^v^=β0+β1ξ+β2η+β3ξη,
where
(47)β0=14(v^1+v^2+v^3+v^4),β1=18(v^1-v^2-v^3+v^4)+34(v^6-v^8),β2=18(v^1+v^2-v^3+v^4)-34(v^5-v^7),β3=14(v^1-v^2+v^3-v^4),
thus, when α=(1,0), we have
(48)D^α(Π~^v^)=β1+β3η.
Obviously, {1,η} is a basis of D^αQ^1. Moreover, let w^=∂v^/∂ξ, we can get
(49)β1=18(v^1-v^2-v^3+v^4)+34(v^6-v^8)=-18∫-11(∂v^∂ξ(ξ,-1)+∂v^∂ξ(ξ,1))dξ+38∫-11(∂v^∂ξ(1,η)+∂v^∂ξ(-1,η))dη=-18∫-11∫-11∂∂η(η∂v^∂ξ)dξdη+38∫-11∫-11∂∂ξ(ξ∂v^∂ξ)dξdη=:F1(w^),β3=14(v^1-v^2+v^3-v^4)=14∫-11(∂v^∂ξ(ξ,1)-∂v^∂ξ(ξ,-1))dξ=14∫-11∫-11∂2v^∂ξ∂ηdξdη=:F2(w^).
By Hölder’s inequality, Fj(w^) (j=1,2) is a bounded linear functional on H1(K^). Therefore, by the basic theorem in [26], (45) holds. Then,
(50)∥(v-Π~v)x∥0,K2=hx-2hxhy∥(v^-Π~^v^)ξ∥0,K^2≤Chx-2hxhy|v^ξ|1,K^2=Chx-2hxhy(|v^ξξ|0,K^2+|v^ξη|0,K^2)=Chx-2(hx4|vxx|0,K2+hx2hy2|vxy|0,K2)≤Ch2|v|2,K2.
The conclusion when α=(0,1) can be derived similarly, that is,
(51)∥(v-Π~v)y∥0,K2≤Ch2|v|2,K2.
Equations (50) and (51) immediately imply the desired result.
The following error estimates can be found in [27].
Lemma 3.
For all u∈H3(Ω), without the regular or quasiuniform assumption, the following estimates hold:
(52)∥u-Πhu¯∥h≤4hπ|u|3,|∑K∈𝒯h∫∂K(Δu-∂2u∂s2)∂vh∂nds|≤Ch|u|3|vh|2,h,∀vh∈Vh,|∑K∈𝒯h∫∂K∂2u∂s∂n∂vh∂sds|≤Ch|u|3|vh|2,h,∀vh∈Vh.
The following theorem shows that for any fixed ɛ∈(0,1] the new incomplete biquadratic element method converges linearly with respect to h.
Theorem 4.
Suppose u and uh are the solutions of (1) and (14), respectively, u∈H3(Ω), then without the quasiuniform assumption and regular condition, there exists a constant C independent of h and ɛ, such that
(53)∥|u-uh|∥ɛ,h≤Ch(ɛ|u|3,Ω+|u|2,Ω).
Proof.
The second Strang lemma implies
(54)∥|u-uh|∥ɛ,h≤C(infvh∈Vh∥|u-uh|∥ɛ,h+supwh∈Vh|Eɛ,h(u,wh)|∥|wh|∥ɛ,h),
where Eɛ,h(u,wh) is the consistency error given by
(55)Eɛ,h(u,wh)=ɛ2ah(u,wh)+b(Πh1u,Πh1wh)-f(Πh1wh).
Furthermore, from Lemmas 1 and 2,
(56)infvh∈Vh∥|u-uh|∥ɛ,h≤∥|u-Πhu¯|∥ɛ,h=(ɛ2|u-Πhu¯|2,h2+|Πh1u-Πh1Πhu¯|1,h2)1/2≤Ch(ɛ|u|3+|u|2).
Hence, it suffices to estimate the consistency error Eɛ,h(u,wh).
Since u∈H3 and Πh1wh are continuous, it follows from (1) that
(57)(f,Πh1wh)=∫ω(-ɛ2D(Δu)+Du)00·D(Πh1wh)dxdy,∀wh∈Vh.
By using the approximation formulation, the consistency error can therefore be expressed as
(58)Eɛ,h(u,wh)=ɛ2∑K∈𝒯h∫K(D2u:D2wh+D(Δu)·DΠh1wh)dxdy+∑K∈𝒯h∫K(DΠh1u-Du)·DΠh1whdxdy.
On the other hand,
(59)D2u:D2wh=ΔuΔwh+(2uxyvxy-uxxvyy-uyyvxx),
and by Green’s formula, Eɛ,h(u,wh) can be rewritten as
(60)Eɛ,h(u,wh)=ɛ2∑K∈𝒯h∫KD(Δu)·D(Πh1wh-wh)dxdy+∑K∈𝒯h∫KD(Πh1u-u)·DΠh1whdxdy+ɛ2∑K∈𝒯h∫∂K(Δu-∂2u∂s2)∂vh∂nds+ɛ2∑K∈𝒯h∫∂K∂2u∂s∂n∂vh∂sds.
It follows from Hölder’s inequality and Lemma 1 that
(61)∑K∈𝒯h∫KD(Δu)·D(Πh1wh-wh)dxdy≤Ch|u|3|wh|2,h,∑K∈𝒯h∫KD(Πh1u-u)·DΠh1whdxdy≤Ch|u|3|Πh1wh|1,h.
Together with Lemma 3, we can immediately get the desired estimate.
Remark 5.
By Lemma 1 and the estimates above, we can derive the same convergence result as in [2]:
(62)ɛ|u-uh|2,h+|u-Πh1uh|1,h≤Ch(ɛ|u|3,Ω+|u|2,Ω).
We should mention that the result here does not need the quasiuniform assumptions. Moreover, similar discussions can also lead to the following estimate when the meshes satisfy the quasiuniform assumption
(63)ɛ|u-uh|2,h+|u-Πh1uh|1,h≤Ch1/2∥f∥0,Ω.
Remark 6.
In the last section, a counterexample is presented to show the possible divergence of incomplete biquadratic plate element when applied to second order problem, but the theorem above implicates that, when the approximation formulation (14) is employed, the uniform convergence result can be ensured even without the regular condition or quasiuniform assumption.
5. Numerical Experiments
In this section, numerical experiments are carried out to confirm our theoretical analysis of the incomplete biquadratic element. We calculate several numerical examples for problem (1) in different approximation schemes. We consider problem (1) with Ω=[0,1]2⊂R2 and f=ɛ2Δ2u-Δu, where u=(sinπxsinπy)2. The domain Ω is divided into the following two fashions.
Mesh 1: square mesh. The mesh obtained in this way for n=16 is illustrated in Figure 3(a).
Mesh 2: each edge of Ω is divided into n segments with n+1 points sin(πi/n)/2, i=0,1,…,n/2,1-sin(πi/n)/2, i=n/2+1,…,n. The mesh obtained in this way for n=16 is illustrated in Figure 3(b).
Mesh fashions: mesh 1 (a) and mesh 2 (b).
We first compute the relative errors in the energy norm ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h under mesh 1 when we use the standard finite element approximation, for different ɛ and h. For a comparison, we also consider the case when ɛ=0, that is, the Poisson equation with Dirichlet boundary conditions, and the biharmonic problem
(64)Δ2u=f,inΩ,u=∂u∂n=0,on∂Ω,
for this case, the corresponding relative error is presented by |u-uh|2,h/|u|3. From Table 1 we can see that if the standard approximation scheme is applied, the method is divergent when ɛ→0.
The errors |∥u-uh∥|ε,h/|∥u∥|ε,h employing approximation form (12) under mesh 1.
ε∖m×n
8×8
16×16
32×32
64×64
20
3.5080e-001
1.8514e-001
9.3821e-002
4.7068e-002
2-2
3.3197e-001
1.7650e-001
8.9611e-002
4.4977e-002
2-4
3.9897e-001
2.3228e-001
1.2090e-001
6.1071e-002
2-6
5.7470e-001
5.3768e-001
3.6322e-001
2.0029e-001
2-8
5.4334e-001
5.7884e-001
6.0446e-001
5.4125e-001
2-10
5.3774e-001
5.5800e-001
5.6811e-001
5.8575e-001
Poisson
5.3734e-001
5.5616e-001
5.6087e-001
5.6205e-001
Biharmonic
4.5852e-002
2.4188e-002
1.2255e-002
6.1481e-003
We first present the relative errors in the energy norm ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h under mesh 1 when we use the standard finite element approximation.
To see the numerical effect more clearly, we plot the errors ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h and ∥u-uh∥0 under different meshes in a logarithm scale in Figure 4. Obviously the slope of the curve represents the convergence rate. We immediately get from the figures that when ɛ<2-8, the errors are no longer descending, which means that the method is divergent. We should point out that, for biharmonic equation, the method is very efficient.
The errors ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h and ∥u-uh∥0 under mesh 1 in the formulation (12).
In Tables 2 and 3, we list the relative errors ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h and the L2 errors ∥u-uh∥0 for the approximation scheme (14) under different meshes. In this case, for biharmonic problem, uh∈Vh is the solution of problem
(65)ah(uh,vh)=(f,Πh1vh),∀vh∈Vh.
The errors |∥u-uh∥|ε,h/|∥u∥|ε,h employing approximation form (14) under mesh 1.
ε∖m×n
8×8
16×16
32×32
64×64
20
3.6561e-001
1.8595e-001
9.3374e-002
4.6737e-002
2-2
3.1147e-001
1.5806e-001
7.9321e-002
3.9697e-002
2-4
1.4057e-001
7.1356e-002
3.5816e-002
1.7925e-002
2-6
8.5681e-002
4.4048e-002
2.2146e-002
1.1088e-002
2-8
8.1275e-002
4.2229e-002
2.1271e-002
1.0642e-002
2-10
8.1006e-002
4.2163e-002
2.1281e-002
1.0662e-002
Poisson
8.0988e-002
4.2159e-002
2.1284e-002
1.0667e-002
Biharmonic
4.8120e-002
2.4478e-002
1.2292e-002
6.1526e-003
The errors |∥u-uh∥|ε,h/|∥u∥|ε,h employing approximation form (14) under mesh 2.
ε∖m×n
8×8
16×16
32×32
64×64
20
3.9326e-001
2.0520e-001
1.0374e-001
5.2017e-002
2-2
3.4184e-001
1.7775e-001
8.9771e-002
4.4999e-002
2-4
1.6659e-001
8.3029e-002
4.1538e-002
2.0775e-002
2-6
7.9112e-002
4.4059e-002
2.2540e-002
1.1326e-002
2-8
7.2810e-002
4.3249e-002
2.2399e-002
1.1253e-002
2-10
7.2776e-002
4.3471e-002
2.2618e-002
1.1408e-002
Poisson
7.2784e-002
4.3503e-002
2.2652e-002
1.1438e-002
Biharmonic
5.1683e-002
2.6973e-002
1.3637e-002
6.8377e-003
When the formulation (14) is employed, the result shows that the method is uniformly convergent with respect to the parameter ɛ. Moreover, we can get that
(66)limɛ→0∥|u-uh|∥ɛ,h∥|u|∥ɛ,h=∥|u-uh|∥0,h∥|u|∥0,h.
At the same time, we plot the logarithm figures of the errors under different meshes in Figure 5. The results consist with our analysis.
The errors ∥|u-uh|∥ɛ,h/∥|u|∥ɛ,h under mesh 1 (a) and mesh 2 (b).
From Figures 5, 6, and 7, we can also see that, with the numerical results derived under mesh 1 and mesh 2 differ slightly, the method under mesh 2 is still very efficient when the approximation scheme (14) is employed. This matches our theoretical analysis.
The error |u-uh|1 under mesh 1 (a) and mesh 2 (b).
The error |u-uh|0 under mesh 1 (a) and mesh 2 (b).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work is supported by the National Natural Sciences Foundation of China (no. 11126272 and 11201122), High-Level Personal Foundation of Henan University of Technology (Grant no. 2009BS066), and The basic and frontier project of Henan province (no. 132300410232).
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