A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming Rotated Q 1 Finite Element Approximation of the Eigenvalue Problems

This paper discusses the nonconforming rotated Q 1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.


Introduction
A posteriori error estimates and adaptive algorithms are the mainstream directions in the study of finite element methods; however, a posteriori error estimates are the theoretical basis of adaptive finite element method.Under these reasons, it is very meaningful to study the a posteriori error estimates.Particularly, it is well known that the residual type a posteriori error estimates usually contain a general constant , which often affects the validity of the error estimates.Then, it is significant that exploring a computable upper bound a posteriori error estimate does not include constant .
The residual type a posteriori error estimate of finite element was first proposed by Babushka and Rheinboldt [1] in 1978 and has been studied and applied to many problems.For example, in 2005, Ainsworth [2] gave the a posteriori error estimate of residual type which can provide a computable upper bound for elliptic boundary value problem.In 2007, based on what Ainsworth researched in [2], Carstensen et al. [3] established a framework of a posteriori error estimates of residual type of a class of nonconforming finite element, which includes the nonconforming - element, the nonconforming rotated  1 element, and Han element, and so forth.In 2010, using the a posteriori error estimates of nonconforming finite element established by Carstensen, Yang [4] founded the a posteriori error indicators for elliptic differential operator eigenvalue problem.Recently, Han and Yang [5] gave a class of a posteriori error estimates of spectral element methods for 2nd-order elliptic eigenvalue problems.
The finite element method is an important approach to solve the Steklov eigenvalue problem (see [6][7][8][9][10]).A posteriori error estimates of finite element for the Steklov eigenvalue problem has attracted attention from mathematical community in recent years.In 2008, Armentano and Padra [11] proposed and analyzed the a posteriori error estimate of the linear finite element approximation for the Steklov eigenvalue problem, and their residual type error estimate can be obtained by the local computation of approximate eigenpairs.In 2011, Ma et al. [12] studied a posteriori error estimate of the nonconforming  rot 1 element for Steklov eigenvalue problem.For the Steklov eigenvalue problems, Yang and Bi [13] have lately obtained the local a priori/a posteriori error estimates of conforming finite elements approximation and Zhang et al. [14] gave certain results of spectral method.The nonconforming rotated  1 element was proposed by Rannacher and Turek [15].Based on the existing research results, we discuss further a computable upper bound a posteriori error estimate of the boundary value problem established by Ainsworth and discover that this error estimate does not include a general constant .So, we use the a posteriori error estimate to establish a computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem.In addition, we extend the error estimate to the Steklov eigenvalue problem, and obtain an efficient computable upper bound a posteriori error indicators.Finally, we verify that the computable upper bound a posteriori error estimate of the boundary value problem is effective (see Table 1).Through calculating the validity of the computable upper bound a posteriori error indicators on Lshaped domain, we can ascertain that the indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective (see Tables 2 and 3).
Let P ℎ be a partition with mesh diameters ℎ of the domain Ω consisting of disjoint convex quadrilateral elements, and the nonempty intersection of any two distinct elements is either a single common node or a common edge.In addition, the nonempty intersection of an element with the exterior boundary is a portion of either Γ  or Γ  .The family of partitions is assumed to be locally quasi-uniform in the sense that the ratio of the diameters of any adjacent elements is bounded above and below uniformly over the whole family of partitions.Define the generalized energy norm |‖V‖| by where the operator grad ℎ satisfies the condition (grad ℎ V)|  = grad(V|  ), ∀ ∈ P ℎ and the notation (⋅, ⋅)  is used to denote the  2 -inner product over a domain .The subscript  is omitted when it is a physical domain .
The nonconforming rotated  1 finite element space (see [15]) is defined by The nonconforming rotated  1 element approximation of (2) is the following: find where

A Posteriori Error Estimate of Boundary Value Problem.
In this subsection we present the computable upper bound a posteriori error estimate of the boundary value problem established by Ainsworth in [2,16].It is the key to establishing a computable upper bound a posteriori error indicator for the eigenvalue problem (1).

Consider the boundary value problem of finding
where The variational form of ( 7) consists of seeking  ∈  1  (Ω) such that The nonconforming rotated  1 finite element approximation of ( 8) is the following: find  ℎ ∈  ℎ, such that To establish a computable upper bound of nonconforming finite element a posteriori estimate for the error  = −  ℎ in the sense of energy norm (3), we use the following Helmholtz decomposition (see [17]) to divide the error  into the conforming part and the nonconforming part.

Lemma 1. Let
where / denotes the tangential derivative in direction .
Then the error  can be decomposed as the form where  ∈  1  (Ω) satisfies and  ∈ H satisfies where curl denotes the operator curl  = (−  ,   ).Moreover, it is valid that Lemma 1 shows that the error  can be decomposed to the conforming part |‖‖| 2 and the nonconforming part The following Theorem 2 gives the error estimate of the conforming part.
Theorem 2. Let  ∈  2 () and  ] ∈  2 () denote the interior residual and the interelement flux jump, respectively.Then where Δ  is a quantity of higher order or even negligible compared with ‖  + (1/2)curl   ‖  −1 , .Both the vector-valued function   and the scalar-valued function   contain the interior residual (see [2]) Moreover, there exists a positive constant , independent of mesh-size, such that for each element there holds where κ is a block including the element  and its adjacent elements.Lemma 3 plays a key role for obtaining the error estimate of the nonconforming part.Lemma 3. Let  ∈ H, H be defined by (10); then Evidently, (18) gives an upper bound of the nonconforming part.It is important to note that the right hand side of ( 18) is the minimum value and the interpolation postprocessing function  * appears in the right hand side of (18).Reference [18] has emphasized that an appropriate selection of  * is the key to obtaining an effective computable upper bound a posteriori error estimate.And this requires that the function  * is of a simple form and computable and makes the error of the nonconforming part effective.
Considering these factors, [2,16] made such selection:  * is taken to be a piecewise (pullback) biquadratic function on each element .The interpolation nodes of the function are the element vertices   , edge midpoints   , and element centers   .The interpolation conditions are given by where P  ⊂ P ℎ denotes the set of elements which share common vertex   , It is obvious that the function  * defined above satisfies  * ∈  1  (Ω) and can be used to obtain an upper bound for the nonconforming part of the a posteriori error estimates.
Theorem 4 gives the reliability and validity of the nonconforming part.Theorem 4. Let  * ∈  1  (Ω) be constructed as described above; then Moreover, there exists a positive constant , independent of any mesh-size, such that Combining ( 14), (15), and (20), we have the following overall a posteriori error estimate: Note that Δ  is a quantity of higher order compared with ‖  + (1/2)curl   ‖  −1 , , or even negligible.Let  ℎ be the approximate solution of (8); we define a computable upper bound a posteriori error indicator by Obviously, the error indicator  com does not include a general constant  and is an effective error indicator (see Table 1).So, we are very interested in the error indicator  com and decide to apply the indicator  com to eigenvalue problem (1).

A Posteriori Error Estimate of the Eigenvalue Problem
In this section, we apply the error indicator  com to the eigenvalue problem (1) and obtain a computable upper bound a posteriori error indicator  2 com with  =  ℎ  ℎ in (16), where ( ℎ ,  ℎ ) is the th eigenpair of (6).
In order to establish the error indicator  2 com , we need the following results, cited from [4,19,20], respectively, as our Lemmas 5, 6, and 7.
Considering the third term of (29), from the interpolation error estimate, we have and that, according to Lemma 5, we know that the second and the third terms are infinitesimals of higher order comparing with the first term.Hence, the error  −  ℎ completely hinges on the first term on the right-hand side of (29); that is, (31) holds.Combining (30) and (31), we obtain (32).
From ( 32) and (30), we can obtain the computable upper bound a posteriori error indicators  2 com and  com for the eigenvalue  ℎ and the associated eigenfunction  ℎ , respectively.

Extension and Application
In this section, we extend the error indicator  com to the Steklov eigenvalue problem and also obtain an effective error indicator  2 com with  =  ℎ  ℎ and  = 1 in ( 16), where  ℎ and  ℎ are the approximations of (37).
The Steklov eigenvalue problem reads as follows: where Ω ⊂  2 is a bounded convex polygonal domain.
The nonconforming finite element approximation of (38) is the following: find where To define two useful operators, we need the source problem (40) associated with (38) and the discrete problem (41).
According to the above consequences, we have the following theorem which can be proved with the approach in [4].
Proof.From the definitions of  and  ℎ , we obtain under the condition of ∫ Ω   = 0, the auxiliary problem exists a unique solution only up to additive constant.Let  be the exact solution and let  ℎ be the approximate solution of ( 52) and ( ℎ ,  ℎ ) be a rotated  1 element eigenpair of (39) obviously,  ℎ =  ℎ .Taking  = − ℎ ,  = 1, and  =  ℎ  ℎ in (7), from the a posteriori error estimate (22) and the definition of  com , we have      ℎ −     ℎ ≤  com .
In Theorem 13, (ℎ 3/2 ) and (‖ ℎ − ‖ −1 ) are generally infinitesimals of higher order comparing with  com .Therefore, we can use  com as a computable upper bound a posteriori error indicator for the eigenfunction  ℎ of (39).
The next corollary gives a relation between the error in the eigenvalue and eigenfunction approximations.

Numerical Examples
This section will report some computational results for the computable upper bound a posteriori error indicators  com and  2 com .For the sake of simplicity, we take  = 1, and the partition P ℎ is uniform square meshes in problems ( 1) and ( 7).We now verify that the error indicator  com is effective for the boundary value problem (7) by the following three different types of test functions.The corresponding boundary conditions are shown in Figures 1, 2, and 3.The numerical results are listed in Table 1.where  ,ℎ ( = 1, 2, 3) denotes the nonconforming rotate  1 element approximations.From Table 1 we find out the ratio  com /‖  −  ,ℎ ‖ ℎ converges that to 1 rapidly, when the number 1 × 2 of the elements increases gradually.Namely, the a posteriori error indicator  com is effective (see Figure 4).
Next we will compute the validity of the error indicator  2 com of the eigenvalue problem (1).The numerical results are listed in Table 2.
In Table 2, we can see that the indicators  2 com for the first eigenvalue and second eigenvalue are effective and reliable, respectively.But the indicator  2 com for the third eigenvalue is distortion, obviously, for which reason is that the eigenfunction is smooth corresponding to the eigenvalue  3 .So, in Theorem 8, the assumptions, in which Ω is a concave domain and the eigenfunction is singular, are necessary.
denotes the a posteriori error indicator of conforming part and  2 nc = |‖ ℎ −  * ‖| 2 the a posteriori error indicator of nonconforming part.Hence, we can use  com as the error estimate indicator of  ℎ .

Table 1 :
The numerical results of Example 1.

Table 2 :
The numerical results of the eigenvalues  1 ,  2 , and  3 .