A TYPE OF FULL MULTIGRID METHOD FOR NON-SELFADJOINT STEKLOV EIGENVALUE PROBLEMS IN INVERSE SCATTERING

. In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.


Finite element method
Now, we introduce the finite element method (cf. [2]) for the non-selfadjoint Steklov eigenvalue problem (2.2) and its corresponding adjoint problem (2.4).
First, we decompose the computing domain Ω ⊂ R ( = 2, 3) into shape-regular triangles or rectangles for = 2 (tetrahedrons or hexahedrons for = 3) and the diameter of a cell ∈ ℎ is denoted by ℎ . The mesh diameter ℎ describes the maximum diameter of all cells ∈ ℎ . Based on the mesh ℎ , we construct the conforming finite element space denoted by ℎ ⊂ . For simplicity, we only consider the linear Lagrange conforming finite element space which is defined as follows where 1 ( ) denotes the space of polynomials of degree ≤ 1. The standard finite element method for (2.2) is to solve the following eigenvalue problem: We give the discretization of the adjoint problem (2.4) in the same finite element space: also we have the relation̂︀ ℎ =̂︀ * ℎ . Hereafter, we use the triple (̂︀ ℎ ,̂︀ ℎ ,̂︀ * ℎ ) to denote the finite element method approximate eigenpair of the non-selfadjoint Steklov eigenvalue problems (2.2) and (2.4).
Then, we introduce the following notation for error estimation Since the ascent of the non-selfadjoint Steklov eigenvalue problem equals to 1, we have the following error estimates. (i) For any eigenfunction approximationŝ︀ ℎ and̂︀ * ℎ of (2.6) and (2.7), respectively, there exist eigenfunctions and * of (2.2) and (2.4), such that

15)
and Here and hereafter is some constant depending on eigenvalue but independent of the mesh size ℎ.

Full multigrid algorithm for non-selfadjoint Steklov eigenvalue problem
In this section, a type of full multigrid method is presented. In order to describe the full multigrid method, we first introduce the sequence of finite element spaces. We generate a coarse mesh with the mesh size and the coarse linear finite element space is defined on the mesh . Then a sequence of triangulations ℎ of Ω ⊂ R is determined as follows. Suppose ℎ1 (produced from by regular refinements) is given and let ℎ be obtained from ℎ −1 via times regular refinements (produce ( ) subelements) such that where the positive number denotes the refinement index and larger than 1 (usually for classical bisection refinement = 2 and = 1). Based on this sequence of meshes, the corresponding nested linear finite element spaces can be built such that The sequence of finite element spaces ℎ1 ⊂ ℎ2 ⊂ · · · ⊂ ℎ and the finite element space have the following relations of approximation accuracy (cf. [8,16]): for = 2, · · · ,

One correction step
First, we present the one correction step to improve the accuracy of the given eigenvalue and eigenfunction approximation. This correction contains solving an auxiliary boundary value problem inexactly on the current finite element space and an eigenvalue problem on a slight extension of the coarsest finite element space. In this paper, we use (̂︀ ℎ ,̂︀ ℎ ,̂︀ * ℎ ) to denote the solution of direct finite element method, see Lemma 2.2. Assume that we have obtained the algebraic eigenpair approximations where (ℓ) denotes the ℓ-th iteration step in the -th level finite element space ℎ . In this subsection, a type of correction step to improve the accuracy of the current eigenpair approximation ( ℎ ) will be given as follows.
Algorithm 3.1. One Correction Step 1. Define the following auxiliary boundary value problems:

9)
where < 1 is a fixed constant independent of the mesh size ℎ and iteration step ℓ. Now, we turn to give the error estimates of Algorithm 3.1, which indicates that the accuracy of numerical eigenpair can be improved after one correction step.
Theorem 3.3. Assume the given eigenpair approximation ( After the One Correction Step defined in Algorithm 3.1, the resultant approximate eigenpair ( ) has the following error estimates Proof. From (2.6) and (3.5) and trace theorem, where is the constant in trace theorem. It leads to the following estimates by using (2.3), (3.10), (3.12) and (3.14) Combining (3.8) and (3.24) leads to the following error estimate for̃︀ Then from (3.24) and (3.25), we have the following inequalities Since ,ℎ ⊂ ℎ , the eigenvalue problem (3.7) can be regarded as a finite dimensional subspace approximation of the eigenvalue problem (2.6). Combining ⊂ ,ℎ and (2.12) the following estimates hold That is the desired result (3.15). Using (2.14), we have the following estimates Similarly, using (2.16), we have If the considering eigenvalue is large or the spectral gap is small, then we need to choose a smaller . Furthermore, we can increase the multigrid steps for boundary value problem to reduce , and then makes and * smaller. However, the practical application is not limited by these requirements. Actually, and the coarsest space only need to match the number of eigenpairs to be computed. In numerical implementations, does not need to be very small (e.g. = √ 2 8 in Subsect. 5.1).

Full multigrid method for non-selfadjoint Steklov eigenvalue problem
Based on the one correction step defined in Algorithm 3.1, a type of full multigrid scheme will be introduced in this subsection. The optimal error estimate with the optimal computational work will be deduced for this type of full multigrid method.
Since the multigrid method for the boundary value problem has the uniform error reduction rate (cf. [20]), we can choose suitable such that < 1 in (3.8) and (3.9). From the definition (3.22) for , it is obvious that < 1 if the mesh size of is small enough. Based on these property, we can design a full multigrid method for non-selfadjoint Steklov eigenvalue problem as follows.

Estimate of the computational work
In this subsection, we turn our attention to the estimate of computational work for the full multigrid method defined in Algorithm 3.2. It will be shown that the full multigrid method makes solving the non-selfadjoint Steklov eigenvalue problem need almost the same work as solving the corresponding linear boundary value problems. Besides, we turn our attention to the estimate of computational work for the full multigrid method defined in Algorithm 3.2.
First, we define the dimension of each level finite element space as := dim ℎ . Then we have
Proof. We use to denote the work involved in each correction step on the -th finite element space ℎ . Based on Algorithms 3.1 and 3.2, Based on the property (3.45), iterating (3.46) leads to This is the desired result and we complete the proof.
Remark 3.9. The high efficiency of the multigrid method for boundary value problems leads to that one does not need to choose large and , please see Section 5 and [20,26]. The computational works ( ) and ( ℎ1 ) for the non-selfadjoint Steklov eigenvalue problem and its adjoint problem depend on the eigenvalue solver. Fortunately, they are very small since the eigenvalue problems which are required to solve are defined on very low dimensional spaces ,ℎ ( = 2, · · · , ) and ℎ1 . Thus, Algorithm 3.2 has the qusi-optimal complexity.

Full multigrid method for computing multiple eigenpairs
Based on full mutigrid method, we can extend Algorithm 3.2 to compute multiple eigenpairs. Firstly, we should introduce the one correction step for computing multiple eigenpairs of non-selfadjoint Steklov problem.
Remark 3.10. We can also obtain the optimal convergence order and almost optimal estimation of computation work of Algorithm 3.4 similar to Theorem 3.6 and Theorem 3.8. For more detail, please refer to [15].

Adaptive full multigrid for multiple non-selfadjoint Steklov eigenvalue problems
In this section, based on the a posteriori error estimators we will establish an adaptive full multigrid for the non-selfadjoint Steklov eigenvalue problem. Here, we only describe the scheme without analysis.
In the above full multigrid method, we refine the mesh uniformly. However, this is not practical since the amount of required memory will increase very rapidly as we refine the mesh. Hence, an efficient refinement strategy is desired. On the one hand, the solution should be resolved well with the refined mesh. On the other hand, the total amount of the mesh elements should be controlled well to make the simulation efficient. Based on the above discussion, adaptive mesh method is a competitive candidate for the refinement strategy.
A standard adaptive mesh process can be described by the following one More precisely, to get ℎ +1 from ℎ , we first solve the discrete equation on ℎ to get the approximate solution and then calculate the a posteriori error estimator on each mesh element. Next, we mark the elements with big errors and these elements are refined in such a way that the triangulation is still shape regular and conforming. Here, we choose the ZZ recovery-based error estimator [45,53] for (2.1). Based on the recovery operator ℎ (cf. [45,53]), for each element ∈ ℎ , we define the local error indicator ℎ ( ℎ , ) and * ℎ ( * ℎ , ) by and the error indicator for a subdomain ⊂ Ω by )︁ 1/2 , and the main error indicator for a subdomain ⊂ Ω bŷ Based on the error indicator (4.2), we choose the Dörfler's marking strategy for approximations { ,ℎ , ,ℎ , * ,ℎ } + −1 = to construct subset ℳ ℎ for local refinement. 1. Given a parameter̂︀ ∈ (0, 1). 2. Construct a minimal subset ℳ ℎ from ℎ by selecting some elements in ℎ such that 3. Mark all the elements in ℳ ℎ .

Numerical results
In this section, some numerical examples are presented to illustrate the efficiency of the full multigrid method proposed in Algorithm 3.2 and AFEM based on full multigrid adaptive finite element method 4.2 for nonselfadjoint Steklov eigenvalue problems, respectively. When ( ) is a real function, (2.2) is a selfadjoint eigenvalue problem. And, we choose ( ) = 4 + 4i in the following examples. In each level of the full multigrid scheme defined in Algorithm 3.2, 3.4 and 4.2, the parameters are set to be = 3 and = 1, respectively. In addition, we take 3 conjugate gradient smooth steps for the presmoothing and postsmoothing iteration step in the multigrid iteration in Step 1 of Algorithm 3.1 and 3.3. 2 ) 2 . Hence = 1. The sequence of linear finite element spaces are constructed on the series of meshes which are produced by the regular refinement with = 2 (producing 2 subelements). In this example, we choose a mesh which is generated by uniform refinement as the initial mesh ℎ1 and the coarsest mesh to produce a sequence of finite element spaces for investigating the convergence behaviors. Figure 1 shows this initial meshes (ℎ 1 = = √ 2/8) Since the exact eigenvalue is unknown, we use the accurate enough approximation [0.686553 + 2.495294i, −0.343047 + 0.850747i, −0.343047 + 0.850747i, −0.950110 + 0.540097i] given by the extrapolation method (see, e.g. [28]) as the first four exact eigenvalues (sorted by real part) to investigate the errors. Algorithm 3.2 is applied to solve (2.1). Figure 2 gives the corresponding numerical results for the first eigenvalue 1 = 0.686553 + 2.495294i. From Figure 2, we find that the full multigrid can obtain the optimal error estimates as the expected one for the direct finite element method, which confirms with the convergence Theorem 3.6 for Algorithm 3.2. Figure 2. The errors of the full multigrid method for the first eigenvalue 1 (left) and first four eigenvalues 1 , · · · , 4 (right) on the square domain for the initial mesh in Figure 1. We also check the convergence behavior for multiple eigenvalue approximations with Algorithm 3.4. Here the first four eigenvalues are investigated. Similarly, we use the same initial mesh shown in Figure 1. The corresponding numerical results are given in Figure 2, which also exhibits the optimal convergence rate of the full multigrid scheme Algorithm 3.4.
It is easy to know that reentrant corners of the dumbbell domain result in the singularities of the eigenfunctions. The convergence order for eigenfunction approximations is less than 1 by the linear finite element method, which is the order predicted by the theory for regular eigenfunctions ( < 1). We consider to use the adaptive Algorithm 4.2 to solve this problem. Figure 4 shows the mesh after 15 adaptive refinements.
by the extrapolation method (see, e.g. [28]) as the first five exact eigenvalues to investigate the errors. First, we investigate the convergent rate of the adaptive posterior error estimatorˆℎ( ℎ , * ℎ , ) ( ⊂ ℎ ) defined in (4.2). Figure 5 presents the corresponding numerical results for the first five eigenfunction approximations. Here, we useˆ, ℎ to denote the -th error estimatorˆℎ( ,ℎ , * ,ℎ , ℎ ). The error estimate of eigenvalues are given in Figure 5, which shows that our multilevel iteration method combines well with the adaptive finite element method naturally and Algorithm 4.2 has the optimal convergence rate.
It is easy to know that the singularities is on the interface. The convergence order for eigenfunction approximations is less than 1 ( < 1) by the linear finite element method, which is the order predicted by the theory for regular eigenfunctions. We also consider to use the adaptive Algorithm 4.2 to solve this problem. Figure 6 shows the mesh after 17 adaptive refinements.
Since the exact solution is unknown, we use the accurate enough approximation [1.451305 − 1.741234i, −0.942417 − 0.542984i, −1.563818 − 0.563167i] given by the extrapolation method (see, e.g. [28]) as the first three exact eigenvalues to investigate the errors. Figure 7 shows the corresponding numerical results by Algorithm 4.2. From Figure 7, we can find that Algorithm 4.2 is able to obtain the optimal error estimate. It shows that Algorithm 4.2 is efficient for solving non-selfadjoint Steklov eigenvalue problems with discontinuous coefficient.

Concluding remarks
In this paper, a type of full multigrid method is designed to solve non-selfadjoint eigenvalue problems based on the multigrid for boundary value problems and the multilevel correction scheme for eigenvalue problems. Furthermore, when the number of smoothing steps is chosen appropriately, our method can reach the optimal convergence rate with the almost optimal computing complexity. At last, we propose a new type of AFEM for multiple eigenvalues based on full multigrid with the almost optimal computing complexity. Three numerical experiments validate the optimality and show that the proposed algorithms can also compute multiple eigenvalues and solve the eigenvalue problems with complex vector.