Research Article | Open Access
Highly Efficient Calculation Schemes of Finite-Element Filter Approach for the Eigenvalue Problem of Electric Field
This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear algebraic system on finer and finer grids. Theoretical analysis and numerical results show that the scheme has high efficiency. Besides, we use interpolation postprocessing technique to improve the accuracy of solutions, and numerical results show that the scheme is an efficient and significant method for eigenvalue problem of electric field.
In recent years, eigenvalue problems of electric field has attracted increasing attention in the fields of physics and mathematics, and its numerical methods (the filter approach, the parameterized approach, and the mixed approach) are also developed further (see [1–7]). Although the filter approach is an effective and important method for solving eigenvalue problems of electric field, its computation costs and accuracy of numerical solutions still need to be improved.
In fact, it is really a challenging job to reduce the computation costs without decreasing the accuracy of finite-element solutions. As we know, two-grid discretization and multigrid discretization are reliable and important methods satisfying the above requirements. Two-grid discretization was first introduced by Xu for nonsymmetric and nonlinear elliptic problems, and so forth (see [8–10]). Later on, it was successfully applied to Stokes equations, semilinear eigenvalue problems and linear eigenvalue problems, and so forth (see [11–16]). Recently, Yang and Bi  established two-grid finite-element discretization and multigrid discretization Schemes based on shifted-inverse power method. Referecnces [6, 7] applied the two Schemes to the mixed approach for eigenvalue problem of electric field, and  applied them to conforming finite element for the Steklov eigenvalue problem. Based on the work mentioned above, this paper discusses two-grid discretization and multigrid discretization Schemes of the filter approach for eigenvalue problem of electric field and analyzes error estimates. They are extensions of Scheme 2 and Scheme 3 in , respectively.
From 1989 to 1991, Lin and Yang firstly pointed out and proved that the function, obtained by using nodes of lower-order element as interpolation nodes to make a higher order interpolation of lower order finite-element solutions, can have global gradient superconvergence. The technique used to obtain global superconvergence was called finite-element interpolation postprocessing or finite-element interpolation correction (see reviews paper  and the references cited therein). For over 20 years, finite element interpolation postprocessing technique has been developed greatly and was applied to a variety of partial differential equations (see [19–25]). It is applied to this paper too. We give Theorem 4.1, and our numerical results show that interpolation postprocessing is an efficient and significant method for solving eigenvalue problems of electric field.
The rest of this paper is organized as follows. In the next section, some preliminaries which are needed are provided. In Section 3, two kinds of finite-element discretization schemes for eigenvalue problem of electric field are given and the error estimates are established. In Section 4, we introduce interpolation postprocessing technique. Finally, numerical experiments are presented.
Let be a bounded polyhedron domain with boundary . We denote by the unit outward normal vector to , by the electric field, and by the time frequency. Let m/s be the light velocity in vacuum, curl operator, and divergence operator.
Consider the following eigenvalue problem of electric field: Let named eigenvalue.
Define function spaces as follows: When is a convex polyhedron, we define the following function space: Denote Let be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition: Set and .
When is a nonconvex polyhedron, let denote a set of edges of reentrant dihedral angles on , and let denote the distance to the set . We introduce a weight function which is a nonnegative smooth function corresponding to . It can be represented by in reentrant edge and angular domain. We shall write . Define the weighted functional spaces: Denote
Note that when is a convex polyhedron, namely, in the case of . Consider the variational formulation: Find with , such that
Let be a regular simplex partition, and let be a space of piecewise polynomial of degree less than or equal to defined on : Then, .
The discrete variational form of (2.8): Find with , such that The eigenpairs of (2.1) must be that of (2.8). But the converse of this statement may not be true, namely, (2.8) has spurious pairs. Hence, (2.10) has spurious pairs.
It is easy to prove that and are symmetric bilinear forms. Next we shall prove that is continuous and -elliptic.
From the definition of , we have Therefore, continuity of is valid. And which indicates that is -elliptic.
Define operator satisfying Define operator satisfying It is easy to prove that , is self-adjoint completely continuous operator, respectively. Actually, for all , we have which shows that is self-adjoint in the sense of inner product . Similarly, we can prove that is self-adjoint in the sense of inner product .
Then, the eigenvalues of (2.8) are sorted as We can construct a complete orthogonal system of by using the eigenfunctions corresponding to : Equation (2.10) has the following equivalent operator form: Then the eigenvalues of (2.10) are sorted as and the corresponding eigenfunctions are where .
In this paper, and , and are all called eigenvalues.
Suppose that the algebraic multiplicity of is equal to . . Let be the space spanned by all eigenfunctions corresponding to of , and let be the space spanned by all eigenfunctions corresponding to all eigenvalues of that converge to . Let , . We also write , , , and .
The Filter Approach
Let be an eigenpair of (2.10), we know that some of these eigenvalues are “real,” but some are spurious (namely, not divergence free). We should filter out the spurious pairs to obtain “real” eigenpairs. Hence, ones designed a filter ratio: The corresponding value of filter ratio is small for “real” pairs since the divergence part of the eigenvector is small, whereas it is large for spurious ones since the curl part small. Noting that when a multiple eigenvalue is dealt with, an additional step must be carried out (see [3, 5]).
Next we introduce error estimates for the filter approach.
Denote Let be orthogonal projection, namely, Then, .
Lemma 2.1. .
Lemma 2.2. Let be the th eigenpair of (2.10) with . Let be the th eigenvalue of (2.8). Then, there exists such that For any , there exists such that where , , and are constants independent of mesh diameter.
In this paper, we will use the following lemma.
Lemma 2.3. Let be an eigenpair of (2.8), then for any , , the Rayleigh quotient satisfies
Proof. The proof is completed by using the same proof steps as that of Lemma 9.1 in .
3. Two-Grid Discretization Scheme and Multigrid Discretization Scheme
Consider (2.19) on (inner product and norm ). We will discuss the high efficiency of two-grid discretization scheme and multigrid discretization scheme next.
Lemma 3.1. For all nonzero ,
Proof. See .
Lemma 3.2. Let be an approximation for , where is not an eigenvalue of , and with . Suppose that , , , , and , satisfy Then where is the separation constant of the eigenvalue .
Proof. See .
3.1. Two-Grid Discretization Scheme
Reference  established the two-grid discretization scheme based on shifted-inverse power method. Next, we will apply the scheme to eigenvalue problem of electric field.
Let and be regular meshes (see ) with diameters and , respectively. Let , , and be a properly small positive number.
Scheme 1. Two-grid Discretization.
Step 1. Solve (2.8) on a coarse grid : Find , such that , and And obtain the “real” eigenpair by filtering process.
Step 2. Solve a linear system on a fine grid : Find , such that And set .
Step 3. Compute the Rayleigh quotient
We use as the approximate eigenpair of (2.1).
Theorem 3.3. Suppose that is properly small. Let be the approximate eigenpair obtained by Scheme 1. Then there exists eigenpair of (2.1), such that where , , and are positive constants independent of mesh diameters, and these constants are decided by (3.11), (3.13), and (3.30) in the following proof.
Proof. We use Lemma 3.2 to complete the proof. Select and . Obviously,
Noting that , for , we have
Combining the above two inequalities with (2.25) and noting that is a small quantity of higher order than , we obtain
From Lemma 2.1, we know that , then there exists a constant independent of , such that
Obviously, there exists , such that
Then, we derive
Hence, by Lemma 3.1, (3.15) and (2.28), we have
Combining the triangle inequality, (2.27) and (3.16), we deduce
Since is small enough and , from (2.26) and (3.17), we know
For , , since is small enough, is the separation constant, we have
From the Step 2 in Scheme 1 and (2.14), we get
Thus and . Note that differs from by only a constant; then Step 2 is equivalent to
From the arguments of (3.9), (3.12), (3.18), (3.19), and (3.22), we see that the conditions of Lemma 3.2 hold. Hence, substituting (3.11) and (3.17) into (3.3), we obtain
Let eigenvectors be an orthogonal basis of (in the sense of inner product ), then
From (3.23), we directly get
From Lemma 2.2, we know that there exist making satisfy (2.26). Let , then
Combining (3.26) and (3.27), we obtain
Besides, by (2.26), we easily know
which together with (3.28) and (2.25) leads to (3.7).
From the continuous embedding of into , we get that there exists a constant independent of meshes, such that Equation (3.7) indicates that converges to in the sense of norm , then converges to in the sense to norm ; thus, . Therefore, when is small enough, we have The proof of Theorem 3.3 is completed.
Let be the smallest singular exponent in the Laplace problem with homogenous Neumann boundary condition, then . Denote .
Corollary 3.4. Suppose that is properly small. Let be an approximate eigenpair obtained by Scheme 1. Then there exists an eigenpair of (2.1), such that when is a convex domain, when is a nonconvex domain, where and are stated in the proof as follows.
Proof. From [1, 4], we know that when is a convex domain, there exists a constant independent of , such that
Substituting the above inequality into (3.7), and noting that is an infinitesimal of lower order comparing with , we know that (3.32) is valid.
And when is a nonconvex domain, there exists a constant independent of , such that where .
Substituting the above inequality into (3.7), we know that (3.34) is valid.
3.2. Multigrid Discretization Scheme
Next, we will discuss finite-element multigrid discretization scheme based on Rayleigh quotient iteration method. Assume that partition satisfies the following condition.
Condition (A). is a family of regular meshes (see ) with diameters , , , , and is a properly small positive number.
Let be the finite-element spaces defined on . Further, let , .
Scheme 2. Multigrid Discretization.
Step 1. Solve (2.8) on a coarse grid : Find , with , such that And obtain the “real” eigenpair by filtering process.
Step 2. , , .
Step 3. Solve a linear system on a fine grid : Find , such that Set .
Step 4. Compute the Rayleigh quotient
Step 5. If , then output , stop. Else, , and return to Step 3.
Proof. We use induction to complete the proof of (3.41).
For , Scheme 2 is actually Scheme 1. Hence, (3.41) is easily obtained from (3.28).
Suppose that (3.41) holds for . Next, we shall prove that (3.41) holds for .
Select , , and . Using the proof method of Theorem 3.3, we deduce Using the triangle inequality and (2.27), we get and together with the induction assumption, yields From Step 3 of Scheme 2, we know that satisfies From the above arguments, we know that the conditions of Lemma 3.2 hold.
Define and as those in Theorem 3.3 (using instead of , instead of ), then where satisfies (2.26). We can derive by Lemma 3.2 and the proof of (3.11) that Substituting (3.44) into the above inequality, we deduce Like the proof method of (3.27), we get From the above two inequalities, we obtain There exists a constant independent of such that Like the proof method of (3.16), we can derive Combining (3.51) and (3.53), we know that (3.41) is valid. Like the proof method of (3.8), we get (3.42), namely, Theorem 3.5 is valid.
Corollary 3.6. Suppose that Condition (A) holds and (namely, ) is properly small. Let be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair of (2.1), such that the following error estimates hold: when is a convex domain, when is a nonconvex domain, where the and are the ones in Corollary 3.4.
4. Interpolation Postprocessing Technique
In this section, we apply interpolation postprocessing technique to the filter approach for eigenvalue problem of electric field.
Let be a regular simplex mesh of . When , the mesh is obtained by dividing each element of the mesh into four congruent triangular elements; when , the mesh is obtained by connecting the midpoints on each edge of the tetrahedral element, which divides each element of tetrahedralization into eight tetrahedral elements.
Let with be a piecewise linear node interpolation operator on . Let with be a piecewise quadratic node interpolation operator on by using the corners of the mesh as interpolation nodes.
Scheme 3. Interpolation Postprocessing Technique.
Step 1. Use linear finite-element filter approach to solve the problem (2.1) on the mesh , and obtain the “real” eigenpair .
Step 2. On , use the value of the function on the corners of the mesh as interpolation conditions to construct a piecewise quadratic interpolation .
Step 3. Compute the Rayleigh quotient:
Here, is the eigenpair corrected.
We develop the work in  to get the following theorem.
Theorem 4.1. Let be an approximate eigenpair obtained by Scheme 3. Assume that and there exists an such that , for some . Then
Remark 4.2. Generally, to 2nd-order elliptic eigenvalue problems, condition is valid (see [18–25]). But to eigenvalue problems of electric field, it is very difficult to prove that . In Section 5, we will verify this theorem by the numerical experiments.
5. Numerical Experiments
In this section, we consider numerical solutions of problem (2.1) on the L-shaped domain and on the square domain . The smallest five exact eigenvalues are , and , , , respectively.
Here the weight is . In the numerical experiments, when is the L-shaped domain, let or ; when is the square domain, we choose . And we use the numerical integral formula with accuracy of order 2 in our experiments.
From the following tables, we know that these three schemes are reliable for solving Maxwell eigenvalue problems. In addition, the accuracy of solutions is improved highly by these schemes.
In Tables 4–6, denote the first five “real” eigenvalues obtained by linear element filter approach directly, denote the first five “real” eigenvalues obtained by Scheme 3, denote the eigenvalues obtained by quadratic element filter approach directly.