Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape

Abstract

Abstract. The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct a computable a priori error estimation for the FEM solution of Poisson's problem, which holds even for nonconvex domains with reentrant corners. Second, new computable lower bounds are developed for the eigenvalues. Because the interval arithmetic is implemented throughout the computation, the desired eigenvalue bounds are expected to be mathematically correct. We illustrate several computation examples, such as the cases of an L-shaped domain and a crack domain, to demonstrate the efficiency and flexibility of the proposed method. 1. Introduction. The eigenvalue problem for the Laplacian, −Δu = λu, has been well investigated in history, mainly in the study of acoustic theory and problems of vibrating elastic membranes and electromagnetic waveguides. There is considerable literature on approximating the eigenvalues by numerical methods; see, e.g., the review papers of Boffi [5] and Kuttler and Sigillito [19] and the lecture book of Weinberger [41].In this paper, we aim to provide concrete lower and upper bounds for eigenvalues of the Laplacian over a polygonal domain. The need for such bounds arises in many practical problems, for example, the estimation of the Poincaré constant in the embedding theorem of Sobolev spaces (see, e.g., [16]) and the investigation of the spectral distribution of the differential operator when exploring the existence of solutions of nonlinear differential equations (see, e.g., [30,27]). All these problems are closely related to the elliptic eigenvalue problem. Moreover, to apply the computational results to a mathematical proof, strictly correct eigenvalue bounds are desired.It is well known that the upper bound for eigenvalues can be easily computed using the Rayleigh-Ritz method, while the lower bound cannot be determined easily. Many theories propose a qualitative analysis of eigenvalue approximation using various numerical methods, but a concrete lower bound is usually unavailable because of the appearance of unknown constants in the estimation formula. In the following, we present a short review of methods that can provide explicit bounds:

Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

“…We finally compare our results with some existing ones from [16,41,39]. In what concerns the unit square and the first eigenvalue of Section 7.1, our estimates appear sharper while comparing Table 2 with the estimates presented in [16], see Figure 6.2 therein.…”

“…In what concerns λ i , a very precise choice is to use λ i := ∇u ih 2 − η 2 i , where η 2 i was first computed with a rather rough bound λ i . For λ i+1 , if the analytic bounds are too rough to be useful, guaranteed and easily computable numerical bounds can be used from Liu and Oishi [41] (on convex domains for d = 2), Carstensen and Gedicke [16], or Liu [39], typically on a quite coarse mesh. Finally, as a "practical gratis" strategy for λ i+1 , one may simply use λ (i+1)h computed by the linear algebra toolbox when solving for (λ ih , u ih ), see, e.g., Saad [53] and the references therein.…”

“…Recently, though, there has been an important progress in obtaining guaranteed lower bounds for the eigenvalues, especially for the first one: Luo et al [42], Hu et al [32,33], Carstensen and Gedicke [16], Yang et al [63], or Liu [39] achieve so via the lowest-order nonconforming finite element method, Kuznetsov and Repin [37] andŠebestová and Vejchodský [54,55] give numerical-method-independent estimates based on flux (functional) estimates, Liu and Oishi [41] elaborate fine a priori approximation estimates for lowestorder conforming finite elements, and, most recently, Xie et al [62] also rely on fluxes. Earlier work comprises Kato [34], Forsythe [24], Weinberger [61], Bazley and Fox [4], Fox and Rheinboldt [25], Moler and Payne [45], Kuttler and Sigillito [35,36], Still [57], Goerisch and He [27], Plum [48], Behnke et al [5], and Armentano and Durán [1], see also the references therein.…”

“…Sometimes, though, restrictions may apply. A condition on relative closeness to the (first) eigenvalue is necessary in [37,Remark 3.2], [54, condition (3.6)], and [55, condition (5.23)] (in these references, the bounds actually do not converge with the correct speed); solution of an auxiliary eigenvalue problem for nonconvex domains is requested [41]; potential overestimation on adaptively generated meshes may hamper the bounds of [41,16,39], relying on a priori estimates and employing the largest mesh element diameter; an auxiliary global flux problem needs to be solved in [62]; a saturation assumption may be necessary, see the discussion in [32].…”

“…This can be guaranteed in many cases of practical interest by a domain inclusion argument Ω − ⊆ Ω ⊆ Ω + with known smaller and larger eigenvalues λ i−1 ≤ λ i−1 (Ω − ) and λ i+1 (Ω + ) ≤ λ i+1 and by requesting λ i−1 =: λ i−1 (Ω − ) < λ ih < λ i+1 := λ i+1 (Ω + ). Numerical bounds λ i−1 ≥ λ i−1 (typically available during the calculation) and λ i+1 ≤ λ i+1 (obtained a on a coarse mesh by the approach of [41,16,39]) can also be used, see Remarks 5.3 and 5.4 below. We also suppose that the approximation spaces consist of appropriate piecewise polynomials.…”