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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 189490, 15 pages
http://dx.doi.org/10.1155/2014/189490
Research Article

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, [18]. 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 [811]).

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, [1316] 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 [1922].

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

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

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

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.