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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 189490, 15 pages

http://dx.doi.org/10.1155/2014/189490

## An Adaptive Nonconforming Finite Element Algorithm for Laplace Eigenvalue Problem

School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China

Received 11 January 2014; Revised 2 April 2014; Accepted 29 April 2014; Published 29 May 2014

Academic Editor: Agacik Zafer

Copyright © 2014 Yuanyuan Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We establish Crouzeix-Raviart element adaptive algorithm based on Rayleigh quotient iteration and give its a priori/a posteriori error estimates. Our algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

#### 1. Introduction

A posteriori error estimates and adaptive methods of finite element approximation for eigenvalue problems are topics attracting more attention from mathematical and physical fields; see, for example, [1–8]. Basically, there are the following three ways of combining adaptivity and eigenvalue problems in which the a posteriori error estimators are more or less the same but different in the problem solved in each iteration: (1) solving the original eigenvalue problem (see Algorithm 10). The convergence and optimality of this adaptive procedure were proved in [2]; (2) inverse iteration type (with or without correction). The convergence has been studied in [1, 6, 7]; (3) Shifted-inverse iteration type (see [8–11]).

The triangular Crouzeix-Raviart element (C-R element) was first introduced by Crouzeix and Raviart [12] in 1973 to solve the stationary Stokes equation. After that, many scholars developed and applied it to eigenvalue problems, for instance, [13–16] discussed a posteriori error estimates and the adaptive methods of the C-R element. C-R element has important properties; for example, Armentano and Durán [17] discovered and proved that the C-R element eigenvalues approximate the exact ones of the Laplace operator from below, which is a very important property in engineering and mechanics computing.

Based on the above work, this paper further discusses the third kind of adaptive methods of the C-R finite element method for eigenvalue problems and obtains the following new results:(1)we establish a multiscale discretization scheme of the C-R element based on Rayleigh quotient iteration and prove its convergence and a priori error estimates;(2)we give residual type a posteriori error estimator for our adaptive algorithm, as well as its reliability and efficiency;(3)we establish an adaptive algorithm (Algorithm 11), which is performed under the package of Chen (see [18]), and satisfactory numerical results are obtained.

As for the fundamental theory of finite elements and spectral approximation, we refer to [19–22].

Throughout this paper, denotes a positive constant independent of mesh parameter, which may not be the same constant in different places. For simplicity, we use the notation to mean that , and to mean that and .

#### 2. Preliminaries

Consider Laplace eigenvalue problem where is a polygonal domain with the maximum interior angle .

We denote the real order Sobolev spaces with norm by , . Let and be the inner product and the norm in the space , respectively.

The weak form of (1) is as follows: find , , such that where

As we know, is a symmetric, continuous, and -elliptic bilinear form on , and is a symmetric, continuous, and positive definite bilinear form on .

Define the operator , satisfying

Then, (2) has the equivalent operator form , where the operators and are self-adjoint and completely continuous.

Let be a regular triangulation of the domain denote the set of all element edges in , denote the set of interior edges, denote the set of all boundary edges, and denote the set of the midside nodes of the edges . For the set of midpoints of the edges , we use the notation ; for any element , we let be the union set of edges of , and let be the diameter of . The domain consists of all elements sharing at least a side with . For any edge , is the length of , and and are unit outward normal vector and unit tangential vector, respectively.

Given a nonnegative integer , the space consists of polynomials of total degree at most defined over . The C-R element space is given by , is continuous at each , and for .

The C-R element approximation of (2) is given as follows: find , , such that where

is a symmetric, continuous, and -elliptic bilinear form on . is well known as the norm of the space ; let which consists of ,

Define the operator , satisfying

Then, (5) has the equivalent operator form , where the operators and are self-adjoint and completely continuous.

Suppose that and are the th eigenvalue of (2) and (5), respectively, and the algebraic multiplicity of is equal to , . Let be the space spanned by all eigenfunctions corresponding to , and let be the direct sum of eigenspaces corresponding to all eigenvalues of (5) that converge to . Let , .

Define

Define , which is the consistent item of nonconforming finite element. Let be an eigenpair of (2), and then

We need the lemmas as follows (see [11, 23]).

Lemma 1 (see [11, Lemma 2.5]). *Let be an eigenpair of (2) and then, for any with , the Rayleigh quotient satisfies
*

*Lemma 2 (see [11, Lemma 2.4] and [23, Lemma 2.3]). Let and be the th eigenvalue of (5) and (2), respectively. Then,
For any eigenfunction corresponding to , satisfying , there exists such that
For any , there exists such that
*

*3. A Priori Error Estimates for Multiscale Discretization Scheme*

*3. A Priori Error Estimates for Multiscale Discretization Scheme*

*In this section, we will discuss a priori error estimates of the C-R finite element multiscale discretization scheme based on the shift-inverse power method. Let be a family of shape-regular meshes and let be the C-R finite element spaces defined on . Besides, let , .*

*The following condition results from [10, 24].*

*Condition 1. *There exists a properly small positive number , , , such that , .

*The following scheme is proposed by Yang and Bi (see [11]).*

*Scheme 3 (multiscale discretization scheme). *Consider the following steps. *Step **1.* Solve (5) on : find such that and
*Step **2.* Execute , , .*Step **3.* Solve a linear system on : find such that
Set .*Step **4.* Compute the Rayleigh quotient
*Step **5.* If , then output , stop. Else, , and return to Step 3.

*Let be the th eigenpair of (16), and then derived from Scheme 3 is the th eigenpair approximation of (5).*

*In the sequel, we also denote , .*

*Lemma 4 (see [11, Lemma 3.1]). For any nonzero ,
Denote .*

*Our analysis is based on the following crucial property of the shifted-inverse iteration in finite element method (see Lemma 4.2 of [24]), which is a development of Theorem 3.2 in [11]. Let , .*

*Lemma 5 (see [24, Lemma 4.2]). Let and be the th eigenvalue of and , respectively, be an approximation for the eigenpair , where is not an eigenvalue of , and with . Suppose that (C1);(C2) for , , where is the separation constant of the eigenvalue ;(C3) satisfy
Then,
*

*Let us construct the interpolation postprocessing operator (see [25]): on the vertex of elements,
where is the number of elements containing the vertex and is the union of elements containing the vertex .*

*Lemma 6. Suppose that Condition 1 holds and is properly small. Let be obtained by Scheme 3 for , and then there exists such that
*

*Proof. *Based on the proof of Theorem 5.1 in [11] and Lemma 5, we deduce that
and thus (23) holds. Using Strang Lemma and Lemma 3.1 of [25], we deduce that
From the above formula and in [11], we get
and thus (24) holds.

*Based on [10, 11, 24], we will prove the following Theorems 7 and 8 for Scheme 3.*

*Theorem 7. Let be an approximate eigenpair obtained by Scheme 3, and and approximate and , respectively, and , . Suppose that is properly small and Condition 1 holds. Then, there exists such that
where .*

*Proof. *Let , . Since , by calculation, we get
From the definition of , it is easy to know that
From , we get
and thus
By Lemma 3.1 in [25], we get that
Thus,
Using the above formula and (31), we can deduce that
Using Lemma 4, we get
Using triangle inequality and (15), we have
From (12), for , , we have
Noticing that is small enough and Condition 1 holds, then by (38) and (39), we can obtain
Since is the separation constant, is small enough, and Condition 1 holds, we have
From the definition of , we can see that Step 3 in Scheme 3 is equivalent to
where ; that is,
Noticing that differs from by only a constant, then Step 3 is equivalent to
From the above formulae, (41), (42), and (45), we can see that the conditions in Lemma 5 hold; therefore, substituting (39) and (40) into (21), we derive
Let eigenfunctions be an orthonormal basis of in the sense of inner product and then
Let
and then it follows directly from (46) that
By Lemma 2, there exists so that satisfies (14).

Let
Then, , and
By calculation,
where .

From (12) and (13), we deduce that
Substituting (52) into (51), we have
Let
By the above two equalities, we obtain
From (49) and (53), we have
Therefore,
By Lemma 1, we have
Since , using Strang Lemma and Lemma 3.1 of [25], we deduce that
Substituting (59) and (61) into (60), (29) holds.

By (56) and triangle inequality, we have
where .

By (57) and (29), we know that ; thus, (28) holds.

By calculation,
By the above formulae and (12), we deduce that
which together with (49) leads to (30). This completes the proof.

*Theorem 8. Let be the th approximate eigenpair of (1) obtained by Scheme 3, let be the th eigenvalue of (1), and let be properly small. Suppose that Condition 1 holds, then there exists such that
*

*Proof. *The proof of (65) is completed by using induction. When , by Lemma 6, we know that Theorem 8 holds.

Suppose that Theorem 8 holds for ; that is,
which together with the assumptions in Theorem 8, we know that Theorem 7 holds. For , by (29) and (59), we get (65). The proof is completed.

*4. A Posteriori Error Estimates for Multiscale Discretization Scheme*

*4. A Posteriori Error Estimates for Multiscale Discretization Scheme**Based on the work of [14, 26], in this section, we will discuss a posteriori error estimates of the C-R element approximation for Laplace eigenvalue problem.*

*Consider the boundary value problem corresponding to (2): find such that
and its C-R element approximation: find such that
*

*Let , be two elements sharing one edge . For any piecewise continuous function , we denote by the jump of across .*

*Let be the solution of (68), be the jump of across along , and be the jump of across along ; let be element residual; that is,
For , define the residual on the element as
and thus, for , the residual sum on is given by
For , define the date oscillation by
where stands for a piecewise polynomial approximation of over .*

*For the boundary value problem (67), Carstensen and Hu [27] have proved the following a posteriori error estimates:
where constant is only dependent on minimum angle of , and if the right-hand side of (67) is a piecewise linear polynomial over , then
*

*Selecting in (67) and (68), then the generalized solution and the nonconforming finite element solution are and , respectively, and the a posteriori error indicator of is , which is defined by (71).*

*Define the element residual and the jump residual and for as follows:
*

*For , define the residual on the element as
For , define the residual sum on as
*

*Theorem 9. Suppose that the conditions in Theorem 7 hold and is a finite element space consisting of piecewise linear polynomials, then there exists a positive constant which is independent of mesh parameter, such that
*

*Proof. *Let , and by calculation
By triangle inequality, we have
From triangle inequality, (69)-(70), and (75)-(76), we deduce that
It is obvious that , and, by the trace theorem (see e.g., [28]) and the inverse estimates, we get
Thus,
Combining (82), (85), and (30), we get
Hence, from Condition 1, we know that is a small quantity of higher order than . Using (81), we obtain that is also a small quantity of higher order than .

Therefore, by (28), (73), (81), and (86), we have
which is (78).

Similarly, by (28) and (74), we get
and thus (79) holds.

By (61) and (28), we get
and, by substituting the above relation into (60), we obtain
which together with (78) yields (80). This completes the proof.

*5. Adaptive Finite Element Algorithm Based on Multiscale Discretizations*

*5. Adaptive Finite Element Algorithm Based on Multiscale Discretizations**As we know, The following Algorithm 10 is fundamental and important; see [14, 16] for its detailed theoretical results.*

*Algorithm 10. *Choose parameter .*Step **1.* Pick any initial mesh with mesh size . *Step **2.* Solve (5) on for discrete solution .*Step **3. *. *Step **4.* Compute the local indicators .*Step **5.* Construct by Marking Strategy and parameter . *Step **6.* Refine to get a new mesh . *Step **7.* Solve (5) on for discrete solution . *Step **8. *, and go to Step 4.

*Marking Strategy *. Give parameter .

*Step **1.* Construct a minimal subset of by selecting some elements in such that

*Step **2.* Mark all the elements .

* and are defined as (76) and (77), respectively, with and replaced by and .*

*We have the following adaptive algorithm on the basis of Scheme 3.*

*Algorithm 11. *Choose parameter . *Step **1.* Pick any initial mesh with mesh size . *Step **2.* Solve (5) on for discrete solution .*Step **3. *, . *Step **4.* Compute the local indicators .*Step **5.* Construct by Marking Strategy and parameter . *Step **6.* Refine to get a new mesh *Step **7.* Find such that
Set and compute the Rayleigh quotient
*Step **8*.