A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming Rotated <svg style="vertical-align:-4.32pt;width:25.5px;" id="M1" height="20.450001" version="1.1" viewBox="0 0 25.5 20.450001" width="25.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,15.012)"><path id="x1D444" d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5
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Liu, Jie; Xia, Tian; Jiang, Wei

doi:https://doi.org/10.1155/2014/891278

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 891278 | https://doi.org/10.1155/2014/891278

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

Jie Liu,¹Tian Xia,¹and Wei Jiang²

Academic Editor: Sotiris Ntouyas

Received15 Jul 2013

Accepted15 Dec 2013

Published28 Jan 2014

Abstract

This paper discusses the nonconforming rotated 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.

1. 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 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 [6–10]). 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 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 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 L-shaped domain, we can ascertain that the indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective (see Tables 2 and 3).

2. Model Problem and Preliminaries

2.1. Model Problem

Consider the following eigenvalue problem: where is a planar polygonal domain with boundary , the disjoint sets and form a partition of the boundary of , and is assumed to be nonnegative. For simplicity, we assume that is piecewise constant on the finite element mesh.

Then (1) can be written in a weak form: to seek with such that where , and , .

Let be a partition with mesh diameters of the domain 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 or . 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 by where the operator satisfies the condition and the notation is used to denote the -inner product over a domain . The subscript is omitted when it is a physical domain .

The nonconforming rotated finite element space (see [15]) is defined by where denotes the jump across an interface and the set of element edges. The subspace of is defined by

The nonconforming rotated element approximation of (2) is the following: find such that where . Define . Evidently, is the norm on .

2.2. 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 such that where , .

The variational form of (7) consists of seeking such that

The nonconforming rotated finite element approximation of (8) is the following: find such that

To establish a computable upper bound of nonconforming finite element a posteriori estimate for the error 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 satisfies and satisfies where denotes the operator . Moreover, it is valid that Lemma 1 shows that the error can be decomposed to the conforming part and the nonconforming part .

The following Theorem 2 gives the error estimate of the conforming part.

Theorem 2. Let and denote the interior residual and the interelement flux jump, respectively. Then where , , , while is a quantity of higher order or even negligible compared with . 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 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 , 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 denotes the set of elements which share common vertex , .

It is obvious that the function defined above satisfies 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 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 is a quantity of higher order compared with , or even negligible. Let be the approximate solution of (8); we define a computable upper bound a posteriori error indicator by in which denotes the a posteriori error indicator of conforming part and the a posteriori error indicator of nonconforming part. Hence, we can use as the error estimate indicator of .

Obviously, the error indicator does not include a general constant and is an effective error indicator (see Table 1). So, we are very interested in the error indicator and decide to apply the indicator to eigenvalue problem (1).

3. A Posteriori Error Estimate of the Eigenvalue Problem

In this section, we apply the error indicator to the eigenvalue problem (1) and obtain a computable upper bound a posteriori error indicator with in (16), where is the th eigenpair of (6).

In order to establish the error indicator , we need the following results, cited from [4, 19, 20], respectively, as our Lemmas 5, 6, and 7.

Lemma 5. Let be the th eigenpair of (6) with , let be the th eigenvalue of (2), and let be the eigenspace corresponding to . Then , and there exists with , such that where is the largest inner angle of with the edges parallel with axis. If then and sufficiently close to , and ; then .

Let be the bounded domain. Define operators satisfies and satisfies

It is easy to know that (2) and (6) have the equivalent operator forms and , respectively. Meanwhile, we have the following estimates.

Lemma 6. Under the assumption of Lemma 5. Moreover, if and there exists a positive constant independent of mesh-size and , such that , , and , then there exists with , such that where , , are infinitesimals of higher order.

Lemma 7. Let and be the solutions of problems (2) and (6), respectively. Then

Under the above preparations, we can obtain the following error estimates for the eigenvalue and eigenfunction of problem (1).

Theorem 8. Let be the th nonconforming rotated element eigenpair of (6) with , and let be the th eigenvalue of (2). Moreover, let be a concave domain and let the eigenfunction be singular. Then we have Thus, where is infinitesimal of higher order compared with .

Proof. Taking , in (8), then and . By (22), we have Combining (33) and (27) (taking ), we get which shows that (30) holds.
In order to prove (31) and (32), we define interpolation operator by Let in Lemma 7; the fourth term on the right-hand side of (29) vanishes (see [21, 22]).
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 completely hinges on the first term on the right-hand side of (29); that is, (31) holds. Combining (30) and (31), we obtain (32).

Remark 9. For the nonconforming rotated finite element, only when would the estimation be valid (see [19]). Therefore, it is necessary to assume that be a concave domain.
From (32) and (30), we can obtain the computable upper bound a posteriori error indicators and for the eigenvalue and the associated eigenfunction , respectively.

4. Extension and Application

In this section, we extend the error indicator to the Steklov eigenvalue problem and also obtain an effective error indicator with and in (16), where and are the approximations of (37).

The Steklov eigenvalue problem reads as follows: where is a bounded convex polygonal domain.

We have Steklov eigenvalue problem in its variational formulation: find with , so that where , , . Clearly is a symmetric, continuous, and -elliptic bilinear form defined on .

The nonconforming finite element approximation of (38) is the following: find with , such that where . Define . Evidently, is the norm on and is uniformly -elliptic. In fact, .

To define two useful operators, we need the source problem (40) associated with (38) and the discrete problem (41).

Find , satisfies and find , such that

Using the source problem (40), we define the operators and : where the symbol “” denotes the restriction to . Bramble and Osborn [8] proved that (38) has the operator form .

Since is uniformly elliptic with respect to , the problem (41) has unique solution. We then define the operators: From [10], (39) has the operator form . and are self-adjoint, completely continuous operators and .

For the Steklov eigenvalue problem, we need the following error estimates (see [10]) and expansion (see [21]) which will be used in our subsequent analysis.

Lemma 10. Let be the th nonconforming rotated element eigenpair of (39) with , and let be the th eigenvalue of (38). Then , and there exists with , such that where is the space spanned by eigenvector corresponding .

Lemma 11. Let be an eigenpair of (37) and let be an eigenpair of (39). Then

According to the above consequences, we have the following theorem which can be proved with the approach in [4].

Theorem 12. Under the assumption of Lemma 10. Then there exists with , such that where .

Proof. From the definitions of and , we obtain Let , combining the triangle inequality, (49), (44), and (45), we deduce That is, (48) is obtained.

Based on (48), we have the following computable upper bound a posteriori error estimate of the eigenfunction of (39).

Theorem 13. Under the assumption of Lemma 10. Then exists with , such that

Proof. Consider the auxiliary problem under the condition of , the auxiliary problem exists a unique solution only up to additive constant. Let be the exact solution and let be the approximate solution of (52) and be a rotated element eigenpair of (39) obviously, . Taking , , and in (7), from the a posteriori error estimate (22) and the definition of , we have
For the source problem (40) and (41), taking , then , . We deduce
Let denote the dual space of with norm Setting , it follows that thus, Further, From (53) and (58), we find Substituting (59) into (48), we obtain (51).

In Theorem 13, and are generally infinitesimals of higher order comparing with . Therefore, we can use as a computable upper bound a posteriori error indicator for the eigenfunction of (39).

The next corollary gives a relation between the error in the eigenvalue and eigenfunction approximations.

Corollary 14. Under the assumption of Lemma 10, we have where is infinitesimal of higher order comparing with .

Proof. We define interpolation operator , such that Taking in Lemma 11, for the fourth term and the third term on the right-hand side of (47), we have (see [10], pp: 2397–2398) For the second term on the right-hand side of (47), by (45), we get Thus, Combining (51) and (64), we obtain (60).

Remark 15. Under certain conditions, we can prove that is a infinitesimal of higher order than . From Lemma 11 of Yang et al. [10], we have and provided that . Similarly, if , is a concave domain and the eigenfunction is singular. Then, may reach convergence rate and may reach . So, the error hinges on . By (60) and (64), we can obtain a computable upper a posteriori error indicator for the eigenvalue . The numerical results (see Table 3) show that this hypothesis is appropriate.

5. Numerical Examples

This section will report some computational results for the computable upper bound a posteriori error indicators and . For the sake of simplicity, we take , and the partition is uniform square meshes in problems (1) and (7).

We now verify that the error indicator 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.

Example 1. Consider the equation on the square domain and the L-shaped domain , respectively.
Test function 1: , corresponding .
Test function 2: , corresponding .
Test function 3: , corresponding , where denotes the nonconforming rotate element approximations. From Table 1 we find out the ratio converges that to 1 rapidly, when the number of the elements increases gradually. Namely, the a posteriori error indicator is effective (see Figure 4).
Next we will compute the validity of the error indicator of the eigenvalue problem (1). The numerical results are listed in Table 2.

Example 2. Consider the eigenvalue problem on the L-shaped domain (see Figure 3). Here we take , , and , respectively. denotes the th approximate eigenvalue.

In Table 2, we can see that the indicators for the first eigenvalue and second eigenvalue are effective and reliable, respectively. But the indicator for the third eigenvalue is distortion, obviously, for which reason is that the eigenfunction is smooth corresponding to the eigenvalue . So, in Theorem 8, the assumptions, in which is a concave domain and the eigenfunction is singular, are necessary.

According to the explanation in Remark 15, we compute the validity of the error indicator of the Steklov eigenvalue problem on the L-shaped domain (see Figure 3).

Example 3. Consider the problems (37), where , , and were taken at the midpoint value of the interval [0.893602779, 0.893736398], [1.688598128, 1.688606742], and [3.21786760, 3.217900202], respectively. Taking into account the possibility that , can be omitted.

We can see that the ratio converges to 1 from Table 3, which proves that our supposition is rational in Remark 15.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11361013) and the Science and Technology Fund of Guizhou Province of China (no. [2011]2111).

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