Eigenvalue inclusions via domain decomposition

We describe a method for the calculation of guaranteed bounds for the K lowest eigenvalues of second-order problems with Neumann boundary conditions. Using P2 approximations for the eigenfunctions and RT1 approximations for the gradients of the eigenfunctions in H(div; ), an error bound for the eigenfunctions is established for weak approximations in H1( ). In addition, the rest of the spectrum will be bounded by a domain decomposition method; step by step the eigenvalue problem is decomposed into simpler geometrical situations, where sufficient information on the spectrum is available. 1. Introduction We aim at methods for the inclusion of eigenvalues for the equations Tu = u, where T is a self-adjoint second-order elliptic operator (e. g. see BEHNKE-GOERISCH [4] and PLUM [16] for reviews on this topic). Here, “inclusion” is meant in the sense that real intervals are computed which contain the desired eigenvalues with strict mathematical guarantee. Such methods therefore aim not only at numerical approximations to eigenvalues, but at rigorous proofs of the existence of eigenvalues within the computed intervals. The most efficient eigenvalue inclusion methods are based on variational characterizations of the eigenvalues, which reduce the task of enclosing the K smallest eigenvalues below the essential spectrum (as long as they exist) to the computation of the eigenvalues of a K K matrix eigenvalue problem, which in turn can be enclosed by means of intervalanalysis and -arithmetic [3], [14]. In this way, for example, the well known Rayleigh-Ritz method (based on a min-max characterization of eigenvalues) yields upper bounds to the K smallest eigenvalues below the essential spectrum, provided they exist. Obtaining lower bounds to them is a much more delicate task. Goerisch has proposed complementary variational principles generating the desired matrix eigenvalue problems after a suitable fixing of certain abstract terms (see [4] for a summary), provided that a (rough) lower bound for the (K + 1)-st eigenvalue is known a priori! This (or a similar) requirement for a priori knowledge on the spectrum is common to all methods providing lower eigenvalue bounds. To overcome this fundamental difficulty, GOERISCH [12] and PLUM [15], [16] have proposed a homotopy method which connects a “base problem” with known spectrum to the given one by a continuous family of eigenvalue problems, such that each eigenvalue (with given index) increases along the homotopy. By computing eigenvalue inclusions for several selected values of the homotopy parameter, and using the inclusions obtained for the 1991 Mathematics Subject Classification. 65N25,65N15. Key words and phrases. Eigenvalue enclosures, homotopy methods. This work was supported by DAAD, the British Council, and in part by the Deutsche Forschungsgemeinschaft. 2 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS previous parameter value as the needed a priori information, the method is finally successful in providing all required bounds. In practical applications, the question of finding a suitable base problem and an appropriate homotopy is of fundamental importance. E.g. in [15], a constant coefficient base problem and a coefficient homotopy is proposed for problems in domains with “simple” geometry. For biharmonic problems, W IENERS [17] introduced a homotopy in the boundary condidions. For Dirichlet eigenvalue problems in “complicated” domains, the domain monotonicity of the eigenvalues can be used to construct a domain deformation homotopy. DAVIES [9] has proposed a “Neumann decoupling” to obtain a base problem for Neumann eigenvalue problems, and a re-joining of the decoupled subdomains in an interface homotopy. Here, we follow these homotopy ideas to obtain eigenvalue inclusions for Neumann eigenvalue problems for the Laplacian on domains with “complicated” topology or on unbounded domains. Up to the authors’ knowledge, no other method exists so far which gives guaranteed eigenvalue inclusions for such kinds of problems. Besides finding an appropriate homotopy, one has to address the question of choosing a suitable realization of certain abstract terms in Goerisch’s approach, and, closely related to that, the choice of appropriate basis functions for setting up the related matrix eigenvalue problems and for approximating the eigenfunctions. A naive choice would result in the necessity of using basis functions in H2; this is very restrictive if the exact eigenfunctions do not have this regularity (e.g., if the domain has reentrant corners), and this results in very poor H2-approximations and thus, in very coarse eigenvalue inclusion intervals. Thus, we consider the eigenvalue problem in weak formulation, and choose H1-basis functions and -approximations and a corresponding setting of Goerisch’s method, in combination with a suitable spectral shift. The paper is organized as follows: in Section 2 we summarize appropriate eigenvalue inclusion methods, formulated for the special purpose of our application. Then, in Section 3 the general idea of eigenvalue homotopies is reviewed, and we introduce the methods of successive domain decomposition; for comparison, we comment on the alternative method of homotopy with respect to boundary conditions. Finally, in Section 4 we discuss a numerical algorithm for a realization of the method and we demonstrate the application to two models. In the first example we compute very close bounds for the first eigenvalue of the Laplacian on an unbounded domain; here, the rough (and only upper) bounds obtained previously in [10] can be improved considerably. Note that, in addition, our method provides also guaranteed lower bounds. In the second example, eigenvalue bounds of a Neumann problem in a three-fold connected domain are computed; here, no purely analytic method is known to obtain any nontrivial lower eigenvalue bound. This example demonstrates the applicability of the eigenvalue inclusion method to problems with “complicated” topology. 2. Eigenvalue inclusions for weak approximations We consider the application of the general eigenvalue inclusion setting for left definite problems by means of complementary variational principles [4] to the Laplace problem. Let R2 be a bounded domain with Lipschitz boundary @ , @ closed and V = fv 2 H1( ) j vj = 0g: The case = ; is included in our considerations. In the following, the inner product in L2( ) or in L2( )2 is denoted by h ; i. EIGENVALUE INCLUSIONS VIA DOMAIN DECOMPOSITION 3 For the eigenvalue problem hru;r i = hu; i; 2 V (1) there exists a complete orthogonal sequence of eigenfunctions ui with eigenvalues 0 0 1 For obtaining eigenvalue inclusions, we have mainly to be concerned with getting lower bounds, since upper bounds are easily obtained by the well-known Rayleigh-Ritz method (see e.g. [4, Th. 14]): Let m 2 N, and let e u0; : : : ; e um 2 V denote linearly independent trial functions. Define the (m+ 1) (m+ 1)-matrices c M := (hre ui;re uji); b N := (he ui; e uji): Then, with 0 1 : : : m denoting the eigenvalues of the matrix eigenvalue problem c Mx = b Nx; x 2 Rm+1 (2) we have i i for i = 0; : : : ; m. Since the matrix eigenvalues 0; : : : ; m can be enclosed by more direct methods combining numerical linear algebra ideas with interval arithmetic (see [3]), the Rayleigh-Ritz method provides a rather direct access to upper eigenvalue bounds. For the computation of lower bounds we first establish a more explicit version of the general Theorem 5 in [4]. Our version (Theorem 2.1) is adapted to our specific problem by a suitable choice of certain abstract terms governing the general theorem. For this purpose, we define a second space W = f 2 H(div; ) j n = 0 on @ n g; where n denotes the outer unit normal vector at the boundary and H(div; ) = fv 2 L2( )2 j div v 2 L2( )g. THEOREM 2.1. For (~ ui; ~ i) 2 V W , i = 0; :::; m, and ; > 0 define the matrices M = hr~ ui;r~ uji+ ( )h~ ui; ~ uji N = hr~ ui;r~ uji+ ( 2 )h~ ui; ~ uji+ 2h~ i; ~ ji+ 2 h~ ui + div~ i; ~ uj + div~ ji inRm+1;m+1. Suppose, that the matrix N is positive definite, and let 0 1 m be the eigenvalues of the eigenvalue problem Mx = Nx; x 2 Rm+1: (3) Then, for all i such that i < 0, the interval [ 1 i ; ) contains at least i+ 1 eigenvalues of the continuous problem (1). PROOF. Let X = L2( )3. Depending on the parameter > 0, we define the bilinear forms B[u; v] = hrv;rwi+ hv; wi; u; v 2 V b((u; v; w); (u0; v0; w0)) = hu; u0i+ hv; v0i+ hw;w0i; (u; v; w); (u0; v0; w0) 2 X; 4 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS and the operators T : V ! X; T (u) = (ru; u); S : W V ! X; S( ; v) = ; 1 (v + div ) : By construction, we have b(S( ; v); T ( )) = hv; i; 2 W; v; 2 V (4) and b(T (v); T ( )) = B[v; ]; v; 2 V: (5) For this setting, Theorem 5 in [4] can be applied directly to the shifted eigenvalue problem B[u; ] = ( + )hu; i; 2 V (6) with the approximations (~ ui; S(~ i; ~ ui)) 2 V X . REMARK 2.2. Note that if for some K 2 N the a priori information K is available, then the left endpoint of the enclosing interval given by Theorem 2.1 is a lower bound for K 1 i (for all i 2 f0; : : : ; mg such that i < 0). If in particular m = K 1 and m < 0, Theorem 2.1 therefore provides lower bounds to the K smallest eigenvalues. The aforementioned a priori information is accessible by homotopy methods discussed in the next subsection. For the simple application to one single eigenvalue (m = 0), the theorem reduces to the following corollary. COROLLARY 2.3. If hr~ ui;r~ uii h~ ui; ~ uii < i+1; (7) we have the lower bound( ) h~ ui; ~ uii hr~ ui;r~ uii b(S(~ i; ~ ui); S(~ i; ~ ui)) h~ ui; ~ uii i: PROOF. Applying the theorem to a single approximation, assumption (7) gives M = hr~ ui;r~ uii+ ( )h~ ui; ~ uii < 0 and, following (4), (5) and (7), N = hr~ ui;r~ uii+ ( 2 )h~ ui; ~ uii+ 2h~ i; ~ ii+ 2 h~ ui + div~ i; ~ ui + div~ ii = B[~ ui; ~ ui] 2 b(S(~ i; ~ ui); T (~ ui)) + 2b(S(~ i; ~ ui); S(~ i; ~ ui)) = b(T (~ ui); T (~ ui)) 2 b(S(~ i; ~ ui); T (~ ui)) + 2b(S(~ i; ~ ui); S(~ i; ~ ui)) = b(T (~ ui) S(~ i; ~ ui); T (~ ui) S(~ i; ~ ui)) > 0: Thus, taking i =M=N in Th. 2.1, we obtain a lower bound 1 M=N = M( ) N N M = ( ) h~ ui; ~ uii hr~ ui;r~ uii b(S(~ i; ~ ui); S(~ i; ~ ui)) h~ ui; ~ uii for i. EIGENVALUE INCLUSIONS VIA DOMAIN DECOMPOSITION 5 REMARK 2.4. For a more general version, for proofs and for many variants of eigenvalue inclusion theorems see [4]; our formulation is adapted to a direct application to our numerical approximation procedure described below, which provides the trial functions e ui and e i needed for our eigenvalue inclusions. In addition, we restrict ourselves to the Laplace problem. Of course, this can easily be extended to more general bilinear forms, but here we want to focus on the domain decomposition homotopy. Moreover, we want to point out that the main difficulty in the application of the general theorem, i.e. the construction of appropriate functions in X , is removed in our Theorem 2.1; observe that our approach is different from the construction in [4], Theorem 6. The required input information for Theorem 2.1 is computed within two steps. To illustrate this, we restrict ourselves to a polygonal domain . For a given triangulation E of into triangles E 2 E we use conforming Lagrange elements of degree k ~ V = f~ v 2 V j ~ vjE 2 Pk for all E 2 Eg; for the approximations e ui of the eigenfunctions, and Raviart-Thomas finite elements ~ W = f~ 2 W j ~ jE 2 P 2 k 1 + Pk 1 xy for all E 2 Eg; for the approximations e i of the gradients. In the applications, we take k = 2. Now, the enclosure algorithm consists of the following steps: 1. Compute numerical approximations (~ i; ~ ui), i = 0; :::; m of the discrete eigenvalue problem in ~ V hr~ ui;r i = ~ ih~ ui; i; 2 ~ V : (8) 2. Compute the mixed solution (~ i; ~ qi) in ~ W ~ Q of the saddle point problem h~ i; i + h~ qi; div i = 0; 2 ~ W; hdiv~ i; pi = h ~ ui; pi; p 2 ~ Q; (9) where the corresponding Lagrange parameter space is ~ Q = f~ q 2 L2( ) j ~ qjE 2 Pk 1 for all E 2 Eg (see [6] for more details on Raviart-Thomas elements). 3. Apply the Rayleigh-Ritz method and Theorem 2.1 with ~ m < m+1 for a small parameter (in the following remark we discuss the special choice = jj~ ui+div~ ijj). If all required quantities are evaluated in interval arithmetic, this gives guaranteed results for the continuous problem. Note that for the application of the theorem it is sufficient to solve the discrete problems approximately; the exact discrete solution does not enter our analysis. For the required information m+1 we introduce a homotopy in the next section. REMARK 2.5. As we will demonstrate in our examples in Section 4, the described method is very efficient, but we cannot prove that the convergence (h ! 0) of the algorithm is of optimal order. For the discussion of the efficiency we consider approximations uh 2 Vh and h 2 Wh for an eigenfunction u 2 V corresponding to a single eigenvalue 6= 0, where h denotes a mesh size parameter. For simplicity, assume jjuhjj = 1 and let h = jjruhjj2. Then, with additional regularity assumptions, for the approximated eigenvalue 6 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS j hj = O(h2k) could be obtained (e. g., see [13], Corollary 11.2.21). Here, we make weaker assumptions on the approximations: jju uhjj+ jjr(u uh)jj = O(hk); jj(1= h)ruh hjj+ jjuh + div hjj = O(hk): Note that this requires the regularity u 2 Hk+1( ). Then, we have j hj = O(hk), = jjuh+div hjj = O(hk), and jj hjj2 = 1= h+O(hk). This gives for the lower bound provided by Corollary 2.2: ( ) huh; uhi hruh;ruhi b(S( h; uh); S( h; uh)) huh; uhi ( ) h (jj hjj2 + (1= )jjuh + div hjj2) 1 O(hk) h ( ) h (1 +O(hk) + h) h O(hk) h h O(hk) h +O(hk) O(hk) = h(1 O(hk)) O(hk): 3. The homotopy method for the computation of spectral bounds The homotopy method will be described using the framework of quadratic forms, cf. [8]. This is applied to eigenvalues on domains which are decomposed into subdomains. The weak continuity will be forced by an additional term in the corresponding bilinear form, which links the eigenvalue problems on the subdomains to the eigenvalue problem on the composed domain via a homotopy. Here, we relax the assumption that is bounded, so that we must take the essential spectrum into account. 3.1. Quadratic forms and eigenvalue homotopies. Let Qt : L2( ) ! [0;1] be a monotonically increasing family of closed quadratic forms for t 2 [0;1], and let Vt = fv 2 L2( ) j Qt(v) <1g be the domain of Qt. The corresponding self-adjoint operators in L2( ) are defined by h(At)1=2v; (At)1=2vi = Qt(v); v 2 Vt (cf. [7, Th. 4.12]), and we denote the eigenvalues of At below inf ess(At) by 0 (t) 0 (t) 1 (t) 2 counted by multiplicity. There may be finitely or infinitely many (or none at all). LEMMA 3.1. Let (t) j < inf ess(At) for t 2 [0;1]. Then, (t) j is monotonically increasing in t. PROOF. For (t) j < inf ess(At), the eigenvalues can be characterized by the Rayleigh extremal values (t) j = min U2Sj max u2U Qt(u) jjujj2 ; where Sj = fU L2( ) j U is subspace of dimension jg, cf. [5]. Since Qt are monotonically increasing, (t) j is monotonically increasing as well. EIGENVALUE INCLUSIONS VIA DOMAIN DECOMPOSITION 7 REMARK 3.2. In our applications we have V1 = fv 2 L2( ) j lim t !1Qt(v) <1g and Q1(v) = lim t !1 Qt(v); v 2 L2( ): Under additional compactness assumptions (which are satisfied, e. g., if is bounded), one can prove lim t !1 (t) j = (1) j (10) (combining results from [8, Th. 1.2.3], [7, Th. 4.32], [7, Cor. 4.27], and [7, Cor. 4.29]). Moreover, a weak convergence of the corresponding eigenfunctions can be achieved. In case of eigenvalue convergence (10) a homotopy method can in principle always be applied for obtaining eigenvalue inclusions. There are situations where the convergence (10) does not take place, which however we will not discuss here. The general ideal of using such homotopies for obtaining eigenvalue bounds is to start it at a “base problem” with known spectrum, to terminate it at the given problem, and to compute eigenvalue inclusions for several values 0 < t1 < t2 < < tL = 1 of the homotopy parameter. Here, for each j = 1; : : : ; L, the a priori information (tj) K needed for the application of Theorem 2.1 (see Remark 2.2) can be obtained from the lower bound computed for (tj 1) K in the previous homotopy step, since (tj) K (tj 1) K due to Lemma 3.1. For more details, see [12], [15], [16]. 3.2. Comparison problems via domain decomposition. We consider the eigenvalue problem (1) (with not necessarily bounded) with Neumann boundary conditions (i. e. = ;) and V = H1( ). We assume that there are (at least) K + 1 discrete eigenvalues below the infimum of the essential spectrum. Now, let be decomposed into two parts = 0 [ 1; 0 \ 1 = ;; 01 = 0 \ 1: We abbreviate the set fu 2 L2( ) j u0 = uj 0 2 H1( 0); u1 = uj 1 2 H1( 1)g by H1( 0) + H1( 1). Assume that we know the eigenvalues of (1) in 0 and 1 with Neumann boundary conditions on 01, i. e. with V = H1( ) replaced by H1( 0) and H1( 1), respectively. We denote the union of these eigenvalues by 0 (0) 0 (0) 1 (0) 2 The corresponding eigenfunctions can be regarded as elements of H1( 0) +H1( 1), by zero extension outside 0 and 1, respectively. Furthermore, we assume that the traces of functions in H1( 0) and H1( 1) are well defined in L2( 01) and that the map [ ] : H1( 0) +H1( 1) ! L2( 01) u 7 ! u1j 01 u0j 01 is continuous (e. g., let 01 be a Lipschitz curve). Then, H1( ) can be characterized by H1( ) = fu 2 H1( 0) +H1( 1) j [u] = 0g: Following [9], for t 2 [0;1), we consider the eigenvalue problem: find u(t) i 6= 0 in H1( 0) +H1( 1) and (t) i such that hru(t) i ;rvi 0 + hru(t) i ;rvi 1 + t h[u(t) i ]; [v]i 01 = (t) i hu(t) i ; vi (11) 8 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS for all v 2 H1( 0) +H1( 1). Finally, let 0 (1) 0 (1) 1 (1) 2 be the eigenvalues of (1) below the essential spectrum (with eigenfunctions in H1( )). LEMMA 3.3. For j K, the eigenvalues (t) j are monotonically increasing. PROOF. Lemma 3.1 can be applied with Qt(u) = jjrujj2 0 + jjrujj2 1 + t jj[u]jj2 01 on Vt = H1( 0) +H1( 1), t <1, and V1 = H1( ). The same arguments apply if parts of @ are fixed by Dirichlet boundary conditions ( 6= ;). Furthermore, the proof is identical for a decomposition of into several subdomains. Alternatively, for several subdomains this argument can be repeated step by step, introducing several homotopies. Then, it may be more convenient to split into more subdomains, with a successive application of Lemma 3.3 for t = 0 and t = 1. Then, no eigenvalue computations for the intermediate problems (0 < t <1) are required. 4. Examples for the domain homotopy The eigenvalue enclosure algorithm is implemented in the software toolbox UG [1, 2] and its finite element library [18]. This allows the solution of the discrete eigenvalue problem (8) and of the mixed problem (9) with optimal complexity using multigrid methods. Furthermore, UG supports the computation fully in parallel. The numerical approximations are computed with a block-vector iteration with Ritz-orthogonalization in every step. The most time consuming part in the algorithm is the computation of the small matrices in Rm+1 required for the inclusion theorems. Here, many inner products must be evaluated within guaranteed close bounds. Then, upper and lower bounds for the eigenvalues of the small system are computed for the resulting interval matrix eigenvalue problems (2) and (3) [3]. This is realized using the interval arithmetic package C-XSC (cf. [14]). 4.1. Application to an acoustic waveguide. Let = ( 1;1) ( 1; 1) n C be an infinite strip with a compact obstacle C ( 1;1) ( 1; 1), where we assume that C is symmetric with respect to reflections at the x-axis and the y-axis, cf. Fig. 1. FIGURE 1. Infinite strip with symmetric obstacle We consider the eigenvalue problem: find (u; ) 2 H1( ) R such that hru;r i = hu; i; 2 H1( ): Note that this implies Neumann boundary conditions on @ . EIGENVALUE INCLUSIONS VIA DOMAIN DECOMPOSITION 9 Spectral approximations for such kinds of problems have been computed by many authors, e.g. by EVANS and LINTON [11], but up to the authors’ knowledge no guaranteed eigenvalue inclusions (except the coarse upper bound computed in [10]) have been obtained before. We look for eigenfunctions which are anti-symmetric with respect to the x-axis and symmetric with respect to the y-axis. Equivalently we can look for eigenfunctions in V := fu 2 H1([( 1; 0) (0; 1)] n C) j u(x; 0) = 0 for all x 2 ( 1; 0)g (which generates Neumann boundary conditions on the whole boundary except the part ( 1; 0) f0g). In this symmetry class there exists an isolated, single eigenvalue 0 < 2=4; the rest of the spectrum is continuous (the interval [ 2=4;1)), see [10] for a discussion of the background of this problem. By the methods described in Sections 2 and 3, lower and upper eigenvalue bounds for this problem can be obtained in the bounded domain L = ( L; 0) (0; 1)nC, with boundary conditions given by the sets VN := fu 2 H1( L) j u(x; 0) = 0 for all x 2 ( L; 0)g (which implies Neumann boundary condition for x = L) and VD := fu 2 VN j u( L; y) = 0 for all y 2 (0; 1)g (i. e., Dirichlet boundary conditions for x = L and Neumann boundary conditions on all other parts of @ L), where we fix a length L > 0 such that C ( L; L) ( 1; 1). LEMMA 4.1. Let u 2 VD such that jjrujj2=jjujj2 < 2=4. Then, we have N 0 D; where N denotes the smallest eigenvalue of the Laplacian in VN , and D in VD, respectively. PROOF. Since each u 2 VD can be extended to a function in V , we get 0 = min u2V; u6=0 hru;rui hu; ui min u2VD; u6=0 hru;rui hu; ui = D: Since VD VN , we have N D jjrujj2=jjujj2 < 2=4: Now define VU := fu 2 H1(( 1; L) (0; 1)) j u(x; 0) = 0 for all x 2 ( 1; L)g: Then, the Laplacian in VU has continuous spectrum [ 2=4;1). Thus, since N < 2 4 ; N is the lowest point of the spectrum of in VU+VN . Therefore, Lemma 3.3 (with K = 0) states that N is smaller than (or equal to) the lowest point of the spectrum of in V . This provides N 0. We give numerical results for C = [ 0:5; 0:5] [ 0:5; 0:5] and L = 10, cf. Fig. 2. An upper bound for D is obtained by the Rayleigh quotient of the computed approximate eigenfunction. To obtain lower bounds for N , Corollary 2.3 can be applied directly with 10 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS FIGURE 2. Coarse grid for the eigenfunction approximation = 2:467 < 2=4: this can be shown with the application of Lemma 3.3 to the homotopy from VL +H1(( 0:5; 0) (0:5; 1)) to VN with K = 1, where VL := fu 2 H1(( L; 0:5) (0; 1)) j u(x; 0) = 0 for all x 2 ( L; 0)g: See Tab. 1 for the inclusion results. number of elements lower bound for N upper bound for D 364 1.6922 1.7823 1456 1.7519 1.7783 5824 1.7689 1.7770 23296 1.7727 1.7765 TABLE 1. Eigenvalue bounds for the isolated eigenvalue 0. 4.2. Application to a domain with holes. We consider the eigenvalue problem with Neumann boundary condition in a domain composed of five quadrilateral domains 0 = (0; 5) (0; 1), 1, 2, 3, and 4, see Fig. 3. The composed domains will be denoted by 01 = int( 0 [ 1), 012 = int( 01 [ 2), 0123 = int( 012 [ 3), and 01234 = int( 0123 [ 4) = . This example shall illustrate that our method is applicable to problems with rather nontrivial topology, where apparently no nontrivial lower eigenvalue bound can be obtained by any other method. For the base problems, the Neumann eigenfunctions and eigenvalues in quadrilaterals (0; a) (0; b) are used: uij = cos((i =a) x) cos((j =b) y); ij = (i2=a2 + j2=b2) 2; i; j = 0; 1; 2; ::: For the composed domains, numerical computations are required. The eigenvalues corresponding to eigenfunctions in V0 = H1( 0) +H1( 1) +H1( 2) +H1( 3) +H1( 4) V1 = H1( 01) +H1( 2) +H1( 3) +H1( 4) V2 = H1( 012) +H1( 3) +H1( 4) V3 = H1( 0123) +H1( 4) V4 = H1( 01234) (with notation analogous to Section 3) are combined from the eigenvalues in the quadrilaterals 0, 1, 2, 3, 4 and from the eigenvalues of the composed domains 01, 012, 0123, 01234. Using the method described in the previous sections, the existence of eigenvalues within the given intervals is proven, see Tab. 2. EIGENVALUE INCLUSIONS VIA DOMAIN DECOMPOSITION 11 0 [ 1 ~ 1 = 0:2932 ~ 2 = 1:2650 ~ 3 = 2:5518 ~ 4 = 3:8739 0 [ 1 [ 2 ~ 1 = 0:3038 ~ 2 = 0:8592 ~ 3 = 2:1306 ~ 4 = 2:9280 0 [ 1 [ 2 [ 3 ~ 1 = 0:2331 ~ 2 = 0:6975 ~ 3 = 2:0856 ~ 4 = 2:2229 0 [ 1 [ 2 [ 3 [ 4 ~ 1 = 0:3082 ~ 2 = 0:6869 ~ 3 = 1:0876 ~ 4 = 1:1767 FIGURE 3. Domain homotopy i V0 V1 V2 V3 V4 0 0 0 0 0 0 1 0 0 0 0 (0.3068,0.3073) 2 0 0 0 (0.2319,0.2324) (0.6828,0.6840) 3 0 0 (0.3025,0.3029) (0.3926,0.3927) (1.0806,1.0838) 4 0 (0.2927,0.2930) (0.3926,0.3927) (0.6930,0.6941) (1.1728,1.1757) 5 (0.3926,0.3927) (0.3926,0.3927) (0.8552,0.8564) (1.5707,1.5708) > 1:57 6 (0.3926,0.3927) (1.2629,1.2642) (1.5707,1.5708) (2.0646,2.0756) 7 (1.5707,1.5708) (1.5707,1.5708) (2.1195,2.1265) (2.2137,2.2255) 8 (1.5707,1.5708) (2.5444,2.5596) (2.9213,2.9366) > 2:92 9 (3.5530,3.5531) > 3:55 > 3:55 TABLE 2. Eigenvalue bounds for the union of domains 12 HENNING BEHNKE, ULRICH MERTINS, MICHAEL PLUM, AND CHRISTIAN WIENERS The homotopy ensures that all lowest eigenvalues are included and a bound for the rest of the spectrum is established, cf. Fig. 5. The numerical computations of the homotopy and the comparison of the shape of the eigenfunctions (see Fig. 4 for contour lines of one homotopy step) indicates that cross points in the eigenvalue homotopy can occur. Note that this does not prove the existence of a cross point, and numerically we cannot distinguish this situation from nearly crossing (i.e. veering) branches, but this example illustrates that here Theorem 2.1 needs to be applied with m 1 for proving guaranteed lower bounds. FIGURE 4. Homotopy from the second eigenfunction in V3 to the first eigenfunction in V4.