A new nonconforming mixed finite element scheme for second order eigenvalue problem
Introduction
In this paper, we discuss the following eigenvalue problem where Ω ⊂ R2 is a bounded convex polygonal domain with Lipschitz continuous boundary ∂Ω.
Eigenvalue problems play a very important role in mathematical physics and engineering technology which have won extensive attention by scholars. For the second order elliptic eigenvalue problem, the standard FEMs were studied in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Where [1], [2], [3], [4], [5] estimated the lower and upper bounds for the eigenvalues; [6], [7], [8] derived asymptotic expansions and extrapolations for the eigenvalues; Dari et al. [9] and the authors in [10], [11], [12] proposed posteriori error estimates and adaptive FEM, respectively; Yang and Bi [13] and Xie [14] discussed the two-grid and multigrid methods, respectively; Lin and Xie [15] introduced a multilevel correction scheme. The standard MFEMs were researched in [16], [17], [18], [19], [20], [21], [22]. In which Mercier et al. [16] analyzed the error estimates for eigenvalue and eigenvectors by mixed and hybrid FEMs; Gardini and Lin and Xie [17], [18] obtained the superconvergence results of order O(h2) for the eigenvector u in L2-norm; Lin and Xie [19] investigated the error estimates for eigenvalue and eigenvectors and derived the asymptotic expansions and extrapolation for eigenvalue; Durán et al. and Jia et al. [20], [21] discussed a posteriori error estimates; Chen et al. [22] gave the two-grid method for MFEMs. However, most of the above MFEMs for elliptic eigenvalue problem focused on the conforming elements without consideration on the superconvergence and extrapolation for nonconforming elements.
Recently, a new mixed variational form for second elliptic problem was constructed in [23], [24]. Compared with the standard MFE scheme, the new mixed formulation avoids the divergence space H(div) which makes the theoretical analysis simpler; and the LBB condition is automatically satisfied when the gradient of approximation space for the original variable is included in the approximation space for the auxiliary variable, which leads to easier choice for the mixed approximation spaces. Subsequently, this method was further applied to different equations (see [25], [26], [27], [28], [29]). For problem (1), Weng et al. [26] provided the convergence for eigenvalue and eigenpair in two-grid new MFE scheme via triangular elements, but it did not cover the superconvergence and extrapolation; Shi et al. [27] studied the new formulation with conforming MFEM, and obtained the error estimates for the eigenpair and extrapolation solution for the eigenvalue without analysis of the superconvergence and numerical experiments.
The main purpose of this article is to apply the new MFE scheme to problem (1) with nonconforming elements. The outline of this paper is organized as follows: In Sections 2 and 3, the optimal error estimates for eigenvalue and eigenpair and the approximation from below for eigenvalue are obtained. In Section 4, the superclose and superconvergence results of order O(h2) are derived for both the original variable u in broken H1-norm and auxiliary variable in L2-norm based on the special property of the nonconforming element (when u ∈ H3(Ω), the consistency error is of order O(h2) which is one order higher than that of the interpolation error O(h)) and the techniques of integral identity and interpolation postprocessing. In Section 5, the extrapolation solution with accuracy O(h3) for eigenvalue is received with the help of the asymptotic expansions. In Section 6, some numerical results are provided to illustrate the validity of the theoretical analysis and effectiveness of the proposed method.
Section snippets
New MFE scheme and convergence analysis
Let Ω be a polygon domain with edges parallel to the coordinate axes, Th be a rectangular subdivision of Ω which need not to satisfy the regular condition [30]. For all K ∈ Th, we denote the barycenter of element K by (xK, yK), the four vertices and four sides are respectively. .
The associated MFE spaces Mh (nonconforming element space [31], [32], [33], [34]) and Hh (the lowest order
The approximation of eigenvalue from below
Lemma 3.1 Let be the solutions of (2) and (3), respectively, for all vh ∈ Mh, we have the expansion as follows
Proof Note that
Adding to both sides of (31), it follows from (3) that
Thus
which ends
Superclose and superconvergence analysis
Theorem 4.1 Let be the solutions of (2) and (3), respectively. If we have the following superclose conclusions
Proof The error equations are derived from (1) and (3) as follows
Let . Taking in (36a) and in (36b), then adding them, we have
By use of (12)
Asymptotic expansions and extrapolation
Lemma 5.1 Let be the solutions of (2) and (3), respectively, there holds
where. Proof The definition of gives that therefore
From (3) and (11), we have
and
Numerical results
Consider the problem (1) with . The exact solutions are
We divide the domain Ω into m × n rectangles (m and n are positive integers), and present the numerical results under regular meshes () and anisotropic meshes () respectively.
From Tables 1 and 2 we find the numerical approximation is lower bound of the exact eigenvalue.
Then, the convergence, superclose, superconvergence and extrapolation results are listed in
Acknowledgments
The research was supported by the National Natural Science Foundation of China (grant no. 11271340,11101381). The authors would like to thank the referee for his valuable suggestions on our manuscript.
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