The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

It is worth noting that some researchers [5,6,17] constructed a series of efficient algorithms to solve PDE eigenvalue problems with angular singularity. For elliptic boundary value problem, Xu and Zhou [18] combined two-grid finite element discretization scheme with the local defect correction to propose a general and powerful parallel-computing technique.
is technique has been used and developed by many scholars, for instance, it can be used to solve Stokes equation (see [19,20]), Especially, Xu and Zhou [21], Dai and Zhou [22], and Bi et al. [23][24][25] developed this method and established local and parallel three-scale finite element discretizations for symmetric elliptic singular eigenvalue problems.
As a matter of fact, due to the influence of Coulomb potentials, the convergence order of three-dimensional numerical methods and the computational efficiency of numerical methods will further deteriorate [26]. erefore, one of the most direct and effective methods is to transform the three-dimensional problem into one-dimensional problem. Inspired by [27][28][29] and others references, it is necessary to further study the high-precision numerical method for singular problems. erefore, in this paper, we turn to discuss finite element multiscale discretization based on local defect correction. We further apply local defectcorrection technique proposed by Xu and Zhou to Schrödinger eigenvalue problems, and our work has the following features. (1) We first extend local and parallel three-scale finite element discretizations for symmetric eigenvalue problems established by Dai and Zhou [22] to solve Schrödinger eigenvalue problem. (2) Based on [23], we establish a new multiscale finite element discretization method by local refinement, and this scheme repeatedly makes defect correction on finer and finer local meshes to make up for accuracy loss caused by abrupt changes of local mesh size in three-scale scheme. (3) For the two-scale algorithms in [8,10], we prove the local error estimates of eigenfunctions. (4) Our scheme is simple and easy to carry out, and theoretical analysis and numerical experiment verify its efficiency to solve the singular Schrödinger eigenvalue problem. e rest of this paper is organized in the following way. in Section 2, we will briefly introduce Schrödinger eigenvalue problem and the associated dimension reduction scheme. In Section 3, we will establish the multiscale finite element method. e error estimates of eigenvalues and eigenfunctions will be studied in Section 4. Several numerical experiments are presented in Section 5 to demonstrate the accuracy and efficiency of our algorithm. Some concluding remarks are given in Section 6.
Assume that π h (Ω) � τ { } is a mesh of Ω with mesh-size function h(x) whose value is the diameter h τ of the element τ containing x, and h(Ω) � max x∈Ω h(x) is the mesh diameter of π h (Ω). We write h(Ω) as h for simplicity. Let V h (Ω) ⊂ C(Ω), defined on π h (Ω), be a space of piecewise polynomials, and V 0 Given G ⊂ Ω, we define π h (G) and V h (G) to be the restriction of π h (Ω) and V h (Ω) to G, respectively, and For any G ⊂ Ω mentioned in this paper, we assume that it aligns with π h (Ω) when necessary.
In this part, C denotes a positive constant independent of h, which may not be the same constant in different places. For simplicity, we use the symbol x ≲ y to mean that x ≤ Cy.
We adopt the following assumptions similar as in [18] for meshes and finite element space.
Mathematical Problems in Engineering e finite element approximation of (7) is given; find Define the solution operator T: Problems (7) and (11) have the equivalent operator forms (14) and (15), respectively: e following regularity assumption is needed in theoretical analysis. For any f ∈ L 2 According to [30] and Section 5.5 in [31], the above assumption is reasonable.
For some G ⊂ Ω, we need the following local regularity assumption.
where C Ω and C G are two priori constants. Define the Ritz projection P h : en, T h � P h T (see [32]). Let M(λ) be the space spanned by all generalized eigenfunctions corresponding to λ of T, M h (λ) be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of T h that converge to λ.

Lemma 1.
Let (λ, u) be an eigenpair of (7). en, for all e a priori error estimates of the finite element approximations (11) can be found in [3,32]. and e authors in [18,33] studied the local behavior of finite element. e following results are given in [18]. then Proof. Let p ≥ 2] − 1 be an integer, and let Choose so we have Since A simple calculation shows that It follows from (26)-(29) that Mathematical Problems in Engineering and thus, Similarly, we can obtain By using (31) and (32), we get from (A0) and (A2) and inverse estimate that is completes the proof.
□ Lemma 4. Suppose that G ⊂⊂ Ω 0 ⊂ Ω. en, the following estimates are valid: Proof. By (15), we obtain By the definitions of T, T h , and P h , we deduce that According to Lemma 3, we have en, by using (14) and (39), we conclude that us, we derive (35) from (39).
be finite element spaces of degree less than or equal to r defined on π H (Ω), π w (Ω), and π h i (Ω i ) l 1 , respectively. Based on algorithm B 0 in [22], we establish the following three-scale discretization scheme.
Step 1: solve (7) on a globally coarse grid π H (Ω); find Step 2: solve two linear boundary value problems on a globally mesoscopic grid π w (Ω); find u w ∈ V 0 w (Ω) such that and then, compute the Rayleigh quotient λ w � a(u w , u w )/b(u w , u w ).
Step 3: solve two linear boundary value problems on a locally fine grid π h 1 (Ω 1 ); find e h 1 ∈ V 0 Step 4: set and compute the Rayleigh quotient: We use (λ w,h 1 , u w,h 1 ) obtained by Scheme 1 as the approximate eigenpair of (7).
It is obvious that (λ w , u w ) in Scheme 1 can be viewed as approximate eigenpairs obtained by the two-grid discretization scheme in [8,10] from π H (Ω) and π w (Ω).
Using Scheme 1, abrupt changes of mesh size will appear near zΩ 1 . Influenced by the technique on the transition layer proposed by [23], we repeatedly use the local defect-correction technique to establish the following multiscale discretization scheme.

Scheme 2. (multiscale discretizations based on local defect correction).
Step 1: the same as that of Step 1 of Scheme 1.
Step 2: the same as that of Step 2 of Scheme 1.
Step 5: solve linear boundary value problems on locally fine grid Step 6: set and compute We use (λ w,h l , u w,h l ) obtained by Scheme 2 as the approximate eigenpair of (7).

Theoretical Analysis
Next, we shall discuss the error estimates of Schemes 1 and 2. In our analysis, we introduce an auxiliary grid π h i (Ω) which is defined globally and denote the corresponding finite element space of degree ≤r by V 0 h i (Ω) (i � 1, 2, . . . , l). We also assume that π h i (Ω i ) and V 0 h i (Ω i ) are the restrictions of π h i (Ω) and a subspace of V 0 h i (Ω) to Ω i , respectively, and For D and Ω i stated at the beginning of Section 3, let G i ⊂ Ω and F ⊂ Ω satisfy D ⊂⊂ F ⊂⊂ G i ⊂⊂ Ω i (i � 1, 2, . . . , l). Theorem 1. Assume that M(λ) ⊂ H r+s (Ω) ∩ H r+1 (Ω/D) and (1 < r + s, 0 ≤ s < 1), and H is properly small. en, there exists u ∈ M(λ) such that Proof. Let u ∈ M(λ) such that u − u H satisfies Lemma 2.

Mathematical Problems in Engineering
Proof. Due to the inequality we shall estimate ‖u w,h l − P h l u‖ a,D , ‖u w,h l − P h l u‖ a,G l \D , and ‖u w,h l − P h l u‖ a,Ω\G l , respectively. First, we proceed to estimate ‖u w,h l − P h l u‖ a,D . From (18), (47), and (48), we derive It is obvious that which together with (61) yields , from the above formula and Lemma 3, we deduce that By calculation, we have Substituting the above relation in (64), we obtain To estimate ‖e h l ‖ b,Ω l , we use the Aubin-Nitsche duality argument. For any given f ∈ L 2 (Ω l ), consider the boundary Let φ be the generalized solution of (67) and φ h l and φ h l− 1 be finite element solutions of (67) in V 0 h l (Ω l ) and V 0 h l− 1 (Ω l ), respectively. en, From (47) and (48), we obtain en, by the definitions of φ, φ h l , and e h l , we deduce that Step 2 of Scheme 2 shows that namely, for l � 1, for l > 1, the above formula follows from (47) and (48). erefore, It is clear that Substituting the above two formulae in (70), we derive us, we obtain Substituting (76) in (66), we obtain e combination of (93), (94), and (99) yields (96). □

Numerical Experiments
We will report some numerical experiments by using linear finite element and quadratic spectral element on uniform meshes. In our numerical experiments, we use Scheme 2 to solve the problem such that Ω i � (− 1, − 1 + (1/2 i ) × (3/2)), i � 0, 1, 2, . . ., and locally fine grids have the same degree of freedom as that of globally mesoscopic grid (see Tables 1-3).
In our experiments, the parameter μ is taken to be 1. We set R � 40 for the eigenvalue problem with V � − (1/r) and l � 1 and R � 15 for the other cases. e coarse mesh size and the mesoscopic mesh size satisfy ω � H 2 which means DOF w � DOF 2 H . We use MATLAB 2011b under the package of Chen (see [34]) to solve the problem, and the numerical results are shown in Tables 1-3. is tables corresponds to the results of different potential energy V. From these tables, we can see that, without increasing degree of freedom on locally fine grids, the first local defect correction can largely improve the accuracy of the eigenvalue, and the local defect corrections that follows can gradually improve the accuracy of the eigenvalue by overcoming the singularity at the origin. Here, we set DOF w � DOF Ω i , i � 1, 2, . . . .
In Figure 2, we also plot the error curve of numerical eigenvalues obtained by multiscale Scheme 2. It can be seen that using coarse finite element space with the mesh size H, the error of the finite element eigenvalues is around poor accuracy 10 − 2 . After performing the two-grid iteration, the error of the finite element eigenvalues can be increased from 10 − 2 to 10 − 5 . e final mutiscale iteration can improve the error up to 10 − 6 ∼ 10 − 7 . ese figures show the accuracy and effectiveness of our numerical scheme.

Conclusion
In this paper, we developed a efficient multiscale finite element method for solving the Schrödinger eigenvalue problem with three-dimensional domain. Our scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. e error estimates of eigenvalues and eigenfunctions are proved. Some numerical examples are presented to verify the effectiveness of our numerical method.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.