A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems
This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid . Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking , and when using the - element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking . Finally, numerical experiments are presented to support the theoretical analysis.
To improve the efficiency of finite element method, Xu introduced a two-scale discretization scheme and applied it to nonsymmetric and nonlinear elliptic problems (see [1–3]). Later on, this scheme attracted the attention of academic circles and has been successfully applied to Stokes equations (see [4–7]), semilinear eigenvalue problems (see ), and linear differential operator eigenvalue problems, and so forth.
Up to now, two kinds of finite element two-scale discretization schemes have been developed for linear differential operator eigenvalue problems. The first kind is established by Xu and Zhou  in 2001, whose idea correlates with the iterative Galerkin method which was established by Lin and Xie  and Sloan , but it bases on the finite element spaces on two different scale grids. The scheme of Xu and Zhou has been applied to the conforming finite element method for electric structure eigenvalue problem (see [12–14]), the conforming/nonconforming finite element method for non-self-adjoint eigenvalue problem (see [15, 16]), the conforming/nonconforming finite element method for Steklov eigenvalue problem (see [17, 18]), the mixed finite element method for Stokes eigenvalue problem, and biharmonic equations eigenvalue problem (see [19, 20]). Another group of two-scale discretization scheme is proposed by Yang and Bi . They established the conforming/nonconforming finite element two-scale discretization scheme based on the shifted-inverse power method.
With two-scale discretization schemes, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid , and the resulting solution still maintains an asymptotically optimal accuracy. Thus, the computational efficiency is improved.
Influenced by the work mentioned above, in this paper we establish a new finite element two-scale discretization scheme for mixed variational formulation of eigenvalue problem and apply it to Stokes eigenvalue problem and eigenvalue problem of electric field. The research of this paper has the following features.(1) Our two-scale discretization scheme is a combinative production of the mixed finite element method and the shifted-inverse power method (see [22, Algorithm 27.2]). Comparing with the scheme in [19, 20], the scheme in this paper is more efficient: the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking when solving Stokes eigenvalue problem and when solving eigenvalue problem of electric field; however, with the scheme in [19, 20] the resulted solution maintains an asymptotically optimal accuracy by taking .(2)The literatures of high-efficient numerical method for eigenvalue problem of electric field are not too many by now, thus they seem to be very valuable. Our two-scale discretization scheme is a new and highly efficient method for eigenvalue problem of electric field.
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 eigenvalue problem mixed variational formulation (2.3)-(2.4) in general form, the finite element two-scale discretization scheme based on the shifted-inverse power method is established, and the validity of this scheme is proved theoretically. In Sections 4 and 5, the scheme established in Section 3 is applied to Stokes eigenvalue problem and eigenvalue problem of electric field, respectively. Finally, numerical experiments are presented in Section 6.
Let , , and be three real Hilbert spaces with inner products and norms , , , , and , , respectively. We suppose that (continuously imbedded), is a symmetric, continuous, and -elliptic bilinear form on , that is, is a continuous bilinear form on , that is,
In scientific and engineering computations, many eigenvalue problems for differential equation have the following 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
As for the mixed finite element method for boundary value problems, Brezzi and Babuska, and others have established a systematic theory. Denote
Theorem 2.1 (Brezzi-Babuska Theorem). Suppose that(1) and are continuous bilinear forms, that is, (2) there exists , such that (3)inf-sup condition: there exists , such that then there exists a unique solution to the problem (2.7)-(2.8), and where constant just depends on and . Furthermore, suppose that(4) there exists a constant , such that (5) discrete inf-sup condition: there exists a constant , such that Then there exists a unique solution to the problem (2.9)-(2.10); moreover, the following error estimate is valid: where just depends on and .
Since is a symmetric, continuous, and -elliptic bilinear form on , is a continuous bilinear form, then conditions (1), (2), and (4) of Brezzi-Babuska Theorem hold. Suppose inf-sup condition and discrete inf-sup condition hold. Then by Brezzi-Babuska Theorem, we know that (2.7)-(2.8) and (2.9)-(2.10) are uniquely solvable for each . Thus we can define the corresponding linear bounded operators:
for all for all , It is obvious that (2.3)-(2.4) have an equivalent operator form: Equations (2.5)-(2.6) have an equivalent operator form: It is easy to prove that are self-adjoint operators. In fact, for all , taking in (2.7)-(2.8) we obtain Exchanging and , we get then It shows that is self-adjoint in the sense of inner product . Analogously, it can be proved that is self-adjoint in the sense of inner product .
Assume that (compact imbedded), then it is easy to prove that the operator 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 .
It is obvious that is an inner product on and are equivalent norms. Let in (2.5); then From (2.6), we get . Then therefore, is a completely normal eigenvector system on in the sense of inner product .
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 the th eigenvalue, too. Denote , , and . Define
Lemma 2.2. Suppose that the conditions of Brezzi-Babuska Theorem hold, and . Let be the th eigenpair of (2.5)-(2.6), , and the th eigenvalue of (2.3)-(2.4). Then , and there exists an eigenfunction corresponding to such that Let ; then there exists such that
Proof. From the spectral approximation theory (see ) we have (2.39).
Let satisfy (2.39), and . Next we will prove that this eigenpair satisfies (2.40)-(2.41). From Brezzi-Babuska Theorem and (2.39), we get Using the triangle inequality and (2.43), we deduce that is, (2.40) is valid.
Using the triangle inequality and (2.44), we get which together with yields (2.41).
Let the eigenfunctions be an orthonormal system of in the sense of inner product . Then, from (2.41) and Lemma 3.1 we know that there exists a basis of satisfying and the following result is valid: For any , we write . By calculation, we get From (2.47), when we have , and thus we get .
Denote , then . From (2.47), we deduce that is, (2.42) is valid. The proof is completed.
For , define the Rayleigh quotient
The following lemma is an extension of [23, Lemma 9.1].
3. The Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems
This paper establishes the following finite element two-scale discretization scheme based on the shifted-inverse power method.
Scheme 1. One has the following.
Step 1. Solve the eigenvalue problem (2.3)-(2.4) on a coarse grid : find , such that
Step 2. Solve a equation on a fine grid : find such that Set .
Step 3. Compute the Rayleigh quotient
Next we will discuss the validity of Scheme 1.
Lemma 3.1. For any nonzero elements ,
Proof. See .
Consider the eigenvalue problem (2.25) on the space .
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 .
Theorem 3.3. Suppose that the conditions of Brezzi-Babuska Theorem hold and . Let be the approximate eigenpair obtained by the two-scale discretization scheme and small properly. Then there exists such that
Proof. We use Lemma 3.2 in the proof.
Select and . Let such that satisfies (2.39) and (2.41). By calculation we deduce thus, using Lemma 3.1, we get Using the triangle inequality and (2.42), we derive From Lemma 2.2 we know ; then When is small enough, noting that , from (3.11) and (3.10) we get Having in mind that we have which together with (3.12), 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 2 of Scheme 1, from the definition of and we have Hence, Step 2 of Scheme 1 is equivalent to , .
From (3.21) we obtain Combining (3.22), (3.18), and (3.20), we get By (3.24), taking in (3.23), we obtain Thus From (3.26) we know that the first term on the left-hand side of (3.23) is equal to ; thus then, using discrete inf-sup condition, we obtain Thus Step 2 of Scheme 1 is equivalent to (3.26), (3.28), and . Noting that differs from by only a constant and denoting , we have By (3.13), (3.15), (3.16), and (3.29), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.11) and (3.12) into (3.6), we obtain Let the eigenfunctions be an orthonormal system of (in the sense of ). Then Let and noting that , from (3.30) we deduce By Lemma 2.2, there exists such that satisfies (2.41). Let then . Using (2.41) we deduce Combining (3.33) with the previous inequality, we have Substituting (3.10) into (3.36), we get (3.7).
We know that from Step 2 of two-scale scheme; then Select . From Lemma 2.3, we get Noting that, for all , we have Since (continuously imbedded), . Then from (3.39) we obtain (3.8).
4. Two-Scale Discretization Scheme for Stokes Eigenvalue Problem
Consider the Stokes eigenvalue problem: where is a polygonal domain in denotes the fluid velocity, and denotes the pressure.
In this paper, we use the symbol to stand for vector function. For the function in , let For vector function , define
Find with such that
Let be two mixed finite element spaces. The mixed finite element form is as follows.
Seek with such that
Denote Let . It is clear that is a norm. Then (4.6)-(4.7) and (4.8)-(4.9) can be written in the forms of (2.3)-(2.4) and (2.5)-(2.6), respectively (we need to add for the vector function, e.g., should be written in the forms of ).
We apply Scheme 1 to the Stokes eigenvalue problem (4.6)-(4.7). Adding the symbol for the vector function we get two-scale discretization scheme of mixed finite element for solving the Stokes eigenvalue problem (4.6)-(4.7), which is still called Scheme 1.
Consider the associated source and approximate source problems.
Find such that
Seek such that
From  we know that (4.11)-(4.12) satisfy conditions (1)–(3) in Brezzi-Babuska Theorem; therefore, there exists a unique solution to the problem (4.11)-(4.12) and the following estimate is valid: Condition (4) in Brezzi-Babuska Theorem holds since and . Suppose that condition (5) in Brezzi-Babuska Theorem (discrete inf-sup condition) is valid, then there exists a unique solution to the problem (4.13)-(4.14), and the following error estimate is valid (see [27–29]):
We assume that the following a prior estimate holds: for any , and where is a number determined by the maximal inner angle of . When , (see ).
Suppose that the following estimate holds: for any , and for any ,
Proof. From Brezzi-Babuska Theorem, (4.17), and (4.18), we deduce
By virtue of Nitsche technique (see ) and (4.17), we derive
Using (4.24), we have
Hence, the conditions in Lemma 2.2 and Theorem 3.3 hold.
From (2.23), we know that From (4.23), (4.18), and (4.17), we have Substituting (4.26) and (4.27) into (3.7), we obtain (4.21). Substituting (4.21) and (4.28) into (3.8), we obtain (4.22).
4.1. Mini Mixed Finite Element
Mini element was established by Arnold et al. in 1984 (see ). Let be a regular triangulation of under the meaning of paper , and For any , let , , and denote barycentric coordinates. Denote , and set
From , we know that Mini element satisfies discrete inf-sup condition. From the interpolation theory in Sobolev space, we conclude that (4.18) is valid. Hence, for satisfying (4.17), Scheme 1 for Mini mixed finite element is effective. Theorem 4.1 is valid.
4.2. - Mixed Finite Element
Let be a regular triangulation of , and is the product of refining in the middle point. The - mixed finite element space is defined by where and are piecewise continuous linear polynomial spaces defined on and , respectively.
From [33, Proposition 3.3], we know that - element satisfies discrete inf-sup condition. By the interpolation theory in Sobolev space, we conclude that (4.18) holds. Therefore, for satisfying (4.17), Scheme 1 for - mixed finite element is effective. Theorem 4.1 is valid.
5. Two-Scale Discretization Scheme for Eigenvalue Problem of Electric Field
Consider the eigenvalue problem of electric field: where is a polyhedron in and is the outward normal unit vector on .
In physics, in the above eigenvalue problem of electric field denotes electric field, denotes the time frequency, and is the speed of the light. Usually we set which is called eigenvalue.
The spaces (curl, ), (curl, ) are defined in the usual way:
On the other hand, when is nonconvex the maximal interior angle belongs to . In this situation the problem is relatively complicated. Let denote a set of reentrant edge with dihedral angles belonging to , and let denote the distance to the set : . We introduce a weight function which is a nonnegative smooth function with respect to . It can be represented by in reentrant edge and angular domain. We write . Define the weight function space: 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 both for non-convex and convex domain. We take for non-convex domain; otherwise, we take .
Let be a regular simplex partition (tetrahedral partition) of with the mesh diameter . Define the - finite element space as follows: