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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 190768, 25 pages
Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China
Received 9 July 2012; Revised 4 September 2012; Accepted 12 September 2012
Academic Editor: Xinan Hao
Copyright © 2012 Yidu Yang 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.
This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.
The finite element method for eigenvalue problem of electric field has become a hot topic in the field of mathematics and physics (see, e.g., [1–7]). This paper discusses high efficient mixed finite element calculation schemes for the eigenvalue problem of electric field.
Kikuchi  introduced the first type of mixed variational formulation for the eigenvalue problem of electric field. Based on this formulation, in  Buffa et al.analyzed the approximation of nodal finite element. Boffi et al.  discussed the second type of mixed variational formulation for the eigenvalue problem of electric field and analyzed approximations of edge element and nodal element. Yang et al.  studied a two-grid discretization scheme of finite element for the first type of mixed variational formulation.
Based on the work mentioned above, in this paper a new multi-grid discretization scheme and an adaptive algorithm are proposed for the first type of mixed variational formulation of eigenvalue problem and applied to the eigenvalue problem of electric field. The main features of the research in this paper are as follows.(1)Our multi-grid discretization scheme and adaptive algorithm, which are the extension of conforming finite element multi-grid discretization scheme (see scheme 3 in  and scheme 1 in ), are a combination of the mixed finite element method and the Rayleigh quotient iteration method (see the algorithm 27.3 in ). With our algorithm one solves an eigenvalue problem on a coarse grid just at the first step and always solves a linear algebraic system on finer and finer grids at each following step. We derive the error estimates for the algorithm and prove that the constants appeared in the error estimates are independent of the iteration degrees. Thus we prove the convergence of iterations.(2)The eigenvalue problem of electric field is so complicated that it is very difficult to obtain local a posteriori error estimates for the eigenfunctions of mixed finite element. As yet, there is no achievement reported in this field. Our adaptive algorithm substitutes the weight method established by Costabel and Dauge (see [3, 11]) for local refinement, which uses as a posteriori error estimator of instead of estimating local a posteriori error for the eigenfunction. And the results are satisfying. (3)We analyze the mixed finite element error for the eigenvalue problem of electric field (see Theorem 2.2 and Theorem 4.2). We refer to  to propose a new proof method which differs from the usual one in .
The rest of this paper is organized as follows. Some preliminaries of finite element approximations for eigenvalue problems which are needed in this paper are provided in the next section. In Section 3, for the first type of mixed variational formulation of eigenvalue problem, the finite element multi-grid discretization scheme and the adaptive algorithm are established and the validity of these schemes is proved theoretically. In Section 4, the multi-grid discretization scheme is applied to the eigenvalue problem of electric field. Finally, numerical experiments are presented in Section 5.
Let , , and be three real Hilbert spaces with inner products and norms , , , , , and , respectively. Suppose that (continuously imbedded), is a symmetric, continuous, and -elliptic bilinear form on , that is, is a continuous bilinear form on , that is, It is obvious that is an inner product on and is an equivalent norm to .
In scientific and engineering computations, many eigenvalue problems have the following first type of mixed variational formulation: find , , such that
In order to solve problem (2.3)-(2.4), one should construct finite element spaces and . Restricting (2.3)-(2.4) on , we get the conforming mixed finite element approximation as follows: find , such that
Consider the associated source and approximate source problems. Given , find satisfying Given , find satisfying Note that the source term is independent of the solution.
As for the mixed finite element method for boundary value problems, Brezzi and Fortinand so forth have established a systematic theory (see ). By Brezzi-Babuska Theorem, we have the following.
Lemma 2.1 (Brezzi-Babuska). Suppose that(C1) (2.1)-(2.2) hold;(C2) inf-sup condition holds, namely, there exists a constant , such that then there exists a unique solution to the problem (2.7) and where just depends on , and . Moreover, suppose; (C3) discrete inf-sup condition holds, namely, there exists a constant independent of , such that then there exists a unique solution to the problem (2.8) and the following error estimate is valid: where just depends on and .
Suppose conditions (C1)–(C3) hold in Lemma 2.1. Then there exist unique solutions to the problem (2.7) and (2.8), respectively. Thus, we can define linear bounded operators as follows: , , Obviously, (2.3)-(2.4) has the following equivalent operator form and (2.5)-(2.6) has the following equivalent operator form
It is easy to verify that are self-adjoint operators in the sense of inner product (see ).
Assume (compactly embedded), then it’s easy to prove that is completely continuous, is completely continuous, and is a finite rank operator. Combining (2.3)-(2.4), (2.5)-(2.6), and the -ellipticity of , we deduce Then from the spectral theory of self-adjoint and completely continuous operator, we know that the eigenvalues of (2.3)-(2.4) can be sorted as and the corresponding eigenfunctions are where .
Denote , . In this paper, and , and are all called eigenvalues.
Let be the th eigenvalue with algebraic multiplicity , . is the space spanned by all eigenfunctions corresponding to of . is the space spanned by all eigenfunctions corresponding to all eigenvalues of that converge to . Let , . We call as the th eigenvalue, too. Denote , , and . Define
Theorem 2.2. Suppose , is symmetric, and the conditions of Lemma 2.1 hold; moreover, for any , Then .
Proof. From , we derive that is a completely continuous operator. It is obvious that is a finite rank operator. From (2.12), (2.26), and (2.27), we deduce It shows that pointwisely converges to . From (2.10) and (2.12) we derive that both and are linear bounded. Hence, from Banach-Steinhaus Theorem, we know that there exists a positive constant independent of , such that Thus, is a bounded set in with respect to the unit ball of . From , we know that is a relatively compact set in , which proves that is collectively compact. From (2.28), we know that , and pointwisely converge to . From , are self-adjoint operators in the sense of inner product . Then by Lemma 3.7 or Table 3.1 in , we get . The proof is completed.
Lemma 2.3. Suppose that the conditions of Theorem 2.2 hold. Let be the th eigenpair of (2.5)-(2.6) with , let be the th eigenvalue of (2.3)-(2.4). Then , and there exists an eigenfunction corresponding to , such that let , then there exists such that where depends on in general, and , , and are constants independent of .
For , , define the Rayleigh quotient
Lemma 2.5. Suppose is an approximation of and , then
3. Mixed Finite Element Multigrid Discretization Scheme Based on the Rayleigh Quotient Iteration
In this section, we develop the work in , noticing that in , when and sholud be modified to their absolute values, respectively, and establish the following mixed finite element multi-grid discretization scheme based on the Rayleigh quotient iteration, and give a rigorous theoretical analysis. Suppose the partition satisfies the following conditions.
Condition 1. is a family of regular meshes (see ) with the mesh diameter and , is arbitrarily chosen, , and .
Let and be finite element spaces on .
Scheme 1. Multigrid Discretization.
Step 2. , , .
Step 3. Solve an equation on : find such that Set , .
Step 4. Compute the Rayleigh quotient
Lemma 3.1. For any nonzero ,
Proof. See .
Consider the eigenvalue problem (2.16) on .
Lemma 3.2. Suppose that and are the th eigenvalue of and , respectively, and is an approximate eigenpair, where is not an eigenvalue of , , , , , and satisfy Then where is the separation constant of the eigenvalue .
Proof. See 
Since the convergence rate of and approximating eigenfunctions is lower than that of and approximating eigenfunctions, respectively, the approximation order of is lower than that of . However, in general, the accuracy order of will not exceed that of ; therefore in the following Theorem 3.3 we assume that the accuracy order of is lower than that of .
Theorem 3.3. Suppose that , is small properly, and Condition 1 holds. Let be the approximate eigenpair obtained by Scheme 1, and let approximate , approximate , and the accuracy order of be lower than that of . Then there exists such that where and are determined, respectively, by (3.10) and (3.14) in the following proof.
Proof. Let , and . Since , by calculation we deduce
From Lemma 2.1, there exists a positive constant depending only on , and such that
By Lemma 3.1, we derive
Using the triangle inequality and (2.33), we deduce
According to the hypotheses of the theorem, we know that and are an infinitesimal of lower order comparing with . Hence, there exists a positive constant independent of such that for we have
Note that is small enough and ; from (3.13) and (3.12), we obtain
Noticing that , we have
which together with (3.14), noting that is an infinitesimal of higher order comparing with , yields
Since is the separation constant, is small enough, and , there holds
For in Step 3 of Scheme 1, from the definition of and (taking ), we have
Hence, when , Step 3 of Scheme 1 is equivalent to the following: find such that
And set , .
From (3.23), we obtain Combining (3.24), (3.20) and (3.22), we get By (3.26) and taking in (3.25), we obtain Thus From (3.28) we know that the first term on the left-hand side of (3.25) is equal to , thus then, using discrete - condition we get Thus Step 3 of Scheme 1 is equivalent to (3.28), (3.30), , and .
Noting that differs from by only a constant and selecting , we have By (3.15), (3.17), (3.18), and (3.31), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.13) and (3.14) into (3.6), we obtain Let the eigenfunctions be an orthonormal system of in the sense of , then Let noting , from (3.32) we deduce By Lemma 2.3, there exist such that satisfy (2.32). Let then . Using (2.32), we deduce Combining (3.35) with the above inequality, we have Substituting (3.12) into (3.38), we get (3.7).
From Step 3 of Scheme 1, we know that , thus Select , , and . From Lemma 2.4, we get Noting that , , and , we have Note . From (3.41) we obtain (3.8).
Theorem 3.4. Suppose . Let be small properly. Let be an approximate eigenpair obtained by Scheme 1 (). Then there exists such that
Proof. Consider Scheme 1. Here , . By Lemma 2.3, we know that approximates , and the accuracy order of is lower than . Hence, for , the conditions of Theorem 3.3 hold. Select in the proof of Theorem 3.3 such that satisfies Lemma 2.3. Then from (2.30), we obtain substituting it into (3.7), we obtain (3.42). From (3.8), we deduce (3.43).
From Theorem 3.4 and Theorem 3.3, we know that () and the convergence rate is high. Thus, we use as a posteriori error indicator of (details can be seen in Remark 4.5). Then we establish the following adaptive algorithm.
Scheme 2 (Adaptive Algorithm). Give an error tolerance and choose the parameter , , , and .
Step 2. , , .
Step 3. Solve an equation on : find such that Let , .
Step 4. Compute the Rayleigh quotient
Step 5. If , then select , , and return to Step 3; else output , stop.
Remark 3.5. In Scheme 2, we use as a posteriori error indicator which is global. In order to cope with difficulties caused by local singularity of a complicated problem in calculation, so far, most algorithms designed a local a posteriori error indicator to establish adaptive algorithm with local mesh refinement (e.g., see [20–22]). However, because the eigenvalue problem of electric field is so complicated, that it is very difficult to obtain a local a posteriori error indicator of eigenfunction. Fortunately, the influence of local singularity can be avoided by using the weight method which is established by Costabel and Dauge to discrete the eigenvalue problem of electric field. And the performance of the weigh method is very good (see [3, 11]). Hence, without local mesh refinement, by using the weight method mentioned above our algorithm can also guarantee its high efficiency.
4. The Eigenvalue Problem of Electric Field
Consider the eigenvalue problem of electric field: where is a polyhedron in , is the unit outward normal to .
Physically denotes the electric field, denotes the time frequency, and is the speed of the light velocity. Usually, let named eigenvalue.
When is a nonconvex polyhedron, the problem is relatively complicated. Let denote a set of edges of reentrant dihedral angles on , and denote the distance to the set : . We introduce a weight function which is a nonnegative smooth function of . It can be represented by in reentrant edge and angular domain. We shall write . Define the weighted functional spaces: Denote Let be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition: From the regularity estimate, we know . Let .
In the following discussion, we will use for both nonconvex and convex domains. We select for non-convex domain and for convex domain.
Let be a regular simplex partition (tetrahedral partition) of with the mesh diameter . Define the finite element space .
Restricting (4.8) on the above-mentioned finite space, we obtain the discrete mixed variational form: find such that
We apply Schemes 1 and 2 to the eigenvalue problem of electric field (4.8). Adding the symbol for the vector function, we get a multi-grid discretization scheme and adaptive algorithm for mixed finite element of the eigenvalue problem of electric field which are still called Schemes 1 and 2.
It is easy to know that and are continuous bilinear forms on and , respectively. . (It is true obviously when is convex; when is non-convex, see .)
Consider the source problem associated with (4.8): find such that
For (4.11) and its conforming finite element approximation, condition (C1) of Lemma 2.1 holds obviously;  has proved that condition (C2) holds; assume that the discrete inf-sup condition (C3) holds, then conditions of Lemma 2.1 hold. Thus we can define operators , and . (4.8) and (4.9) can be written as (2.15) and (2.16)-(2.17), respectively.
Theorem 4.2. Suppose that the discrete inf-sup condition (C3) holds. Then . Let be the th eigenpair of (4.9) with , and let be the th eigenvalue of (4.8). Then , and there exists an eigenfunction corresponding to with , such that let , then there exists such that where , , and are constants independent of .
Proof. From the preceding discussion and hypotheses of the theorem, we know that , is symmetric, and the conditions of Lemma 2.1 hold. Besides, since is a regular partition, when is a convex polyhedron, (); when is a non-convex polyhedron, is dense in . Since is dense in , thus, no matter is convex or non-convex, is dense in . For any given , we have . Thus for any , according to the density, we know that there exists such that Selecting being small properly, when , we have where is an interpolation operator. Thus Namely, . Hence (2.26) is true. Analogously, using the density of in , we deduce that , namely, (2.27), is true. Hence, from Theorem 2.2 and Lemma 2.3, we know that , , (4.13), and (4.14) hold. From (2.38), (4.13), and Lemma 4.1, we get (4.12). The proof is completed.
Denote From Lemma 2.1, noting , we deduce
Proof. We use induction to complete the proof. Note that the conditions of Theorem 2.2 hold. For , Scheme 1 is actually two-grid discretization scheme. Substituting (4.19) into (4.12) and (4.13), we derive
Combining (3.7) with and the above two inequalities, we know that there exists such that
Since , substituting (4.23) into (3.43), we deduce
The above two inequalities show that Theorem 4.2 is true for .
Suppose that the theorem is true for , then by Theorem 3.3, we get That is, (4.20) is valid.
From (4.20) and (3.8), we obtain (4.21). The proof is completed.
Assume that is a regular simplex partition (tetrahedral partition) of with the mesh diameter . Let and be the - finite element spaces as follows: Here we set . means that is equal to on the tetrahedron where reentrant edge and angular point are adjacent. Considering finite element approximation of (4.11), for the 3- -iso- Taylor-Hood finite element, Ciarlet and Girault  have discussed that the discrete inf-sup condition (C3) holds when is a convex domain; for the - element, Ciarlet and Hechme have proved that the discrete inf-sup condition (C3) holds when is a polyhedron (see Section 2.2 in  and pp. 509 in ).
Let be the smallest singular exponent in the Laplace problem with homogenous Neumman boundary condition, then . Denote .
Corollary 4.4. Assume that is small properly, , and . Let be an approximate eigenpair of the - element obtained by Scheme 1. Then when is a convex domain, there exists such that When is a non-convex domain, there exists such that where and are determined by (4.31) and (4.32) in the proof, respectively.
Proof. The hypotheses of Corollary 4.4 imply that the conditions of Theorem 4.3 hold. When is convex, for any we have (see in ). Thus there exists independent of such that
Substituting the above inequality into (4.20) and (4.21), we get (4.27) and (4.28), respectively.
When is a non-convex domain, for any , by in  we know that there exists independent of , such that Substituting the above inequality into (4.20) and (4.21), we derive (4.29) and (4.30), respectively.
Remark 4.5. From Corollary 4.4, we see that the constants in the error estimates are not only independent of the mesh diameter but also independent of the iterative degrees. Thus, when , we have . Suppose that the precision order of (4.28) is optimal which cannot be improved any more, then where for convex domain , while for non-convex domain but approximates arbitrarily. Therefore we have that