Convergence Analysis of Incomplete Biquadratic Rectangular Element for Fourth-Order Singular Perturbation Problem on Anisotropic Meshes

and Applied Analysis 3


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][2][3][4][5]).Here, we consider the following two-dimensional linear stationary Cahn-Hilliard type equation as our model problem: where Δ is the standard Laplace operator, Ω is a bounded polygonal domain in  2 , Ω is the boundary of Ω, and  is a real parameter such that 0 <  ≤ 1.Let / denote the normal derivative of  along the boundary Ω.
Particularly, the differential equations (1) formally degenerate to Poisson equations (a plate model degenerates towards an elastic membrane problem) when  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  > 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 fourthorder problems are discretized by a finite element method, the standard variational formulation will require the piecewise smooth functions in  1 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  1 -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.
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  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.
For every V ∈  3 (Ω), we define the interpolation operator Π ℎ as Π ℎ V|  = Π  V, and Π  satisfies such that It can be checked that Let  ℎ be the associated finite element space defined by where [V/] is the jump value of V/ on  ⊂  and Then the corresponding finite element approximation of (4) is as follows: find  ℎ ∈  ℎ , such that where for all  ℎ , V ℎ ∈  ℎ , Next, we will present the modified discretization form of problem (4) in [2].
Let Π 1 ℎ be the interpolation operator of the Lagrange bilinear rectangular element corresponding to the triangulation T ℎ .The modified finite element method of (4) reads as: find  ℎ ∈  ℎ , such that Note that the problem has a unique solution when  > 0, but when  = 0, the problem degenerates to in this case, Π 1 ℎ  ℎ is uniquely determined, though the solution  ℎ is not unique.
We introduce the same mesh dependent norm ‖ ⋅ ‖ ,ℎ and semi-norm | ⋅ | ,ℎ on space  ℎ +   : The energy norm is defined by

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 .That means if the element is applied to a nearly second order problem with the form of (1) when  → 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 to simplify some calculations.Assume Ω = Γ  ∪ Γ  , where Γ  and Γ  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 Let T ℎ be a triangulation of Ω, and let Ṽℎ 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  ℎ ∈ Ṽℎ , such that where ⟨, V ℎ ⟩ = ∫ Γ  V ℎ ,  denotes the arc length along Γ  .Moreover, the exact solution  of (18) satisfies where Let ‖| ⋅ |‖ ℎ be the corresponding energy norm of problem (18), that is, ‖| ⋅ |‖ ℎ =  ℎ (⋅, ⋅).Then, employing Cauchy-Schwarz inequality (we refer to [3]), we have In the following, we will choose a suitable exact solution  to prove the divergence of the method by virtue of (23).The domain Ω is taken as the unit square.To simplify the analytic process, the uniform triangulation with the mesh size 1/ is employed.Assume Γ  to be the intersection of Ω with the coordinate axis, while Γ  to be the part of Ω on  = 1 and  = 1.Hence, the functions in finite element space Ṽℎ are zeros at the vertices on the coordinate axis.
We assume that the exact solution of ( 18) is given by  = .Thus,  =  on  = 1 and  =  on  = 1.Obviously,  is harmonic, which means  = 0. Therefore: Note that  ∈ Ṽℎ , thus Π ℎ  =  and here Π ℎ is the finite element interpolation on Ṽℎ .Similar to the discussion in [3], it is easy to derive that finite element space Ṽℎ can be naturally decomposed into two spaces: where ṼV ℎ and Ṽ ℎ correspond to the vertex values and the edge values, respectively.In fact, the two spaces can be expressed as where E ℎ and X ℎ represent the sets of the edges and vertices corresponding to the triangulation T ℎ , respectively.Let  V ℎ be the interpolant of  onto ṼV ℎ , then, according to its definition, we can get We begin to prove that the limit lim ℎ → 0 (| ℎ (, V ℎ )|/ ‖|V ℎ |‖ 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 Ṽℎ 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  ∈ T ℎ , the length of each edge is 2ℎ = 1/.Let the center point of  be ( 0 ,  0 ), then by the definition of the space ṼV ℎ , direct calculation implies the expression of  V ℎ on element : where We denote the element [−1, 1] × [−1, 1] on (, ) plane by the reference element K. Apparently, the mapping from K to  is affine, hence we have moreover, / = , / = , and then On the other hand, denoting the edges of Ω on  = 1 and  = 1 by Γ  1 and Γ  2 , respectively, we have We first consider the first term, let those components of E ℎ lying on Γ  1 be   ,  = 1, 2, . . ., , and let the corresponding element be   , the center point of   be (  ,  * ), where   = (2 − 1)/2 = ℎ(2 − 1),  * = (2 − 1)/2 = 1 − ℎ.Then Therefore, by a limit process, we can obtain Similarly, for ∫ Γ  2  V ℎ , we have this immediately leads to we can verify that From this expression we obtain that lim This together with (32), (36), and (38) implies The divergence of the method is therefore a consequence of the basic lower bound (23).

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  under anisotropic meshes.
To begin with, we introduce the following error estimate regarding the operator Π 1 ℎ on anisotropic meshes, as to its proof, we refer to [25].

Lemma 1. For the bilinear interpolation operator Π 1
ℎ , for all  ∈  2 (Ω), there holds where  is a constant independent of triangulation.
Denote the quadratic part of the interpolant function ΠV by ΠV and the corresponding part of the function Π  V on the element  by Π  V, then we can get the following lemma.
Lemma 2. For any  ∈ T ℎ , for all V ∈  2 (), without the regular or quasiuniform assumption, we have the following estimate: Proof.Apparently, Π 1  Π  V can be considered as an interpolant of V on  1 ().For simplicity, we denote it by ΠV.We first show that on the reference element K, for multi-index For the convenience of notations, we denote degrees-offreedom by Hence, for all V ∈  1 ( K), direct computation provides the expression of ΠV: where thus, when  = (1, 0), we have Obviously, {1, } is a basis of D Q1 .Moreover, let ŵ = V/, we can get By Hölder's inequality,   (ŵ) ( = 1, 2) is a bounded linear functional on  1 ( K).Therefore, by the basic theorem in [26], (45) holds.Then, The conclusion when  = (0, 1) can be derived similarly, that is, Equations ( 50) and (51) immediately imply the desired result.
The following error estimates can be found in [27].
Lemma 3.For all  ∈  3 (Ω), without the regular or quasiuniform assumption, the following estimates hold: The following theorem shows that for any fixed  ∈ (0, 1] the new incomplete biquadratic element method converges linearly with respect to ℎ. Theorem 4. Suppose  and  ℎ are the solutions of (1) and (14), respectively,  ∈  3 (Ω), then without the quasiuniform assumption and regular condition, there exists a constant  independent of ℎ and , such that          −  ℎ        ,ℎ ≤ ℎ (|| where  ,ℎ (,  ℎ ) is the consistency error given by Furthermore, from Lemmas 1 and 2, inf Hence, it suffices to estimate the consistency error  ,ℎ (,  ℎ ).
Since  ∈  3 and Π 1 ℎ  ℎ are continuous, it follows from (1) that By using the approximation formulation, the consistency error can therefore be expressed as On the other hand, and by Green's formula,  ,ℎ (,  ℎ ) can be rewritten as It follows from Hölder's inequality and Lemma 1 that 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]: 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 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.

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 ⊂  2 and  =  2 Δ 2  − Δ, where  = (sin  sin ) 2 .The domain Ω is divided into the following two fashions.
Mesh 1: square mesh.The mesh obtained in this way for  = 16 is illustrated in Figure 3(a).
We first compute the relative errors in the energy norm ‖| −  ℎ |‖ ,ℎ /‖||‖ ,ℎ under mesh 1 when we use the standard      a logarithm scale in Figure 4. Obviously the slope of the curve represents the convergence rate.We immediately get from the figures that when  < 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.

Figure 1 :
Figure 1: The degrees-of-freedom of incomplete biquadratic element.