van Keer, R.

Vanmaele, M.

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On an internal approximation of a class of elliptic eigenvalue problems

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7M. Vanmaele; R. KeerOn an internal approximation of a class of elliptic eigenvalue problemsIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 301--304.Persistent URL: http://dml.cz/dmlcz/702398Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czON AN INTERNAL APPROXIMATION OF A CLASS OF ELLIPTIC EIGENVALUE PROBLEMS V A N M A E L E M . , V A N K E E R R., G E N T , Belg ium 1 Introduct ion . Let V and H be two real infinite dimensional Hilbert spaces with V compactly and densely embedded in H. Let a : V xV —• R be a bilinear form which is symmetric, bounded and strongly coercive. Let (•,•) be the inner product in H, with norm | • |, Let \\ be a finite dimensional subspace of V . Finally, let (•, -)h> as an approximation of (•, •), be an inner product in \\. With these da ta we introduce the 'solution operators' T : H - V V/ G H,Vv G V : a(Tj\ v) = (j\v) (1.1) fh : Vh - Vh, V / G VhlVvh G Vh : a(fhf\vh) = ( / , vh)h and we consider the corresponding 'exact' and 'approximate' eigenvalue problems (EVP) : Find p G R and u G V : Tu = fi.u Find jj.h G R and uh G Vh : thuh = jxhuh. The former is the operator version of the EVP for a(-, •) in V x V, relative to (•. •), while die latter is equivalent to the EVP for a(-, •) in Vh x Vh relative to (•, -)/i-This paper mainly deals withythe convergence for h —* 0 of an approximate eigenpair, allowing for a multiple exact eigenvalue, under the following hypotheses, met in practice, (|| • || is the norm in V), (HO VveV:mf{\\v-vh\\;vheVh}^0 if h -> 0 (H2) Vvh,wh G Vh : \(wh,vh) - (wh,vh)h\ = \E(wh,vh)\ < e ( H | K | | . | N | , e (h) -> 0 if h->0. (Hi) is the standard approximation property of the finite element siibspa* c> »[' the Sobolev spaces, used in weak variational EVP's for PDE's. In that context, (H2) holds foi (•.)/» corresixMiding to a suitable numerical quadrature for (•,•). In the case of a simple exact eigenvalue the results are incorporated in those of [3], the latter however being obtained in a less transparent manner. Moreover, the present approach can readily be extended to the case that also a(-, •) is approximated suitably on \\ x \\. We rely on [4] (Section V.4.3). First we recall a classical result : Tr = T|v, (1.1). is a compact, self-adjoint, positive definite operator in V. Hence sp(Tr) consists of an infinite sequence of eigenvalues, all being strictly positive and having finite multiplicity, with zero as accumulation 301 point, fii > fi2 > ... > fin > . - . — • 0 (here every eigenvalue occurs as many times as given by its multiplicity). In what follows, C is a generic constant, only depending on V, H and a(-, •)• 2 Uniform convergence of fh to Tr. Let Vh denote the orthogonal complement of Vh in V relative to a(-, •). One easily proves Propos i t ion 2 .1 fh : V - V, fhv = ffct; if v e Vh, fhv = 0 if v e VhL (2.1) defines a compact, self-adjoint, positive operator in V (equipped with a(-,-)j\ having the same eigenpairs as Th, apart from the trivial eigenvalue zero. To 'compare' this extension Th with T, we use the 'intermediate' operator Th : V - V,, V/ 6 V,Vufc e V, : ci(Thf,vh) = (/,u f c). Note that T^ = 7r,vTr, where irh : V —+ Vh is the projection operator relative to a(-,-). Invoking (Hi) and the compactness of Tr, one has, using [1] (Theorem 3.2 p. 124) Propos i t ion 2.2 ||T„ - I * | | S 8 u p { | | ( T - T » H I ; » e v . l H l < 1} -» 0 if h - 0. (2.2) T h e o r e m 2 .1 ||T, - T * | | H sup{| | (T - f*)» | | ;» 6 V, | |v | | < 1} -» 0 if /» - 0. Proof By (2.2) it is sufficient to consider (TA — Th). Denoting by a the coercivity constant of a, one has Vv € V, a| |(T / l - fefc)u||2 < (v - whv,(Th - fh)v) + E(irhvATh - t / J )v ) . Using (H i) , the continuity of i : V —• II and the coercivity and boundedness of a, one gets | | ( T - - 7 * ) i ; | l < C . [ | r - n « | + e(fc).||t-l|]. From a variant of the Aubin-Nitsche lemma, cfr. [1] (Lemma 4.26 p. 215), one finds (with H the unit ball in H) \v - W>| < C.\\v - irhv\\. sup{\\w - T T ^ H ; w £ T(H)} < e(h).\\v\\. Invoking the compactness of i and the spectral decomposition theorem of a compact operator, T, (1.1), may be shown to be compact . (.Hi) then implies that e(h) —• 0 if h —• 0. I 302 3 Convergence of the eigenvalues. Relying on [4] (Section VA.3) one readily obtains Lemma 3.1 Let y. be an eigenvalue of Tr with multiplicity m and isolation distance d. If It is sufficiently small, then the open interval (u - d/2,n-\- d/2) contains exactly m eigenvalues oft'1, counting with their multiplicity. Lemma 3.2 supinf \vh - v\ and supinf \vh -v\< \\Tr - fh\\ , »h V U Vh ' where v and vh run over sp(T r) and sp(Th) respectively. We number the nonzero eigenvalues //J1, 1 < / < dimF^ of t h , similarly to those of Tr. Then, combining the two lemmas, we arrive at Theorem 3.1 \ti - /i/| < ||T r - fefc||, 1 < I < dim Vh, h sufficiently small. Consequently, from Theorem 2.1, fih —> pj, I > 1, if h —> 0. 4 Convergence of the eigenfunctions. Let*//.*-! < Hk = t-*+i = . . . = AU-+m < pA+m+i, i.e. let fik be an {m + l)-fold <-igeuvalue of TP. Denote by uk+r, 0 < r < m, eigenfunctions of T r, cori-esponding to f.tk, urtlumorinal in 11. Let E be the space spanned by these eigenfunctions. Likewise, let t*J+r, 0 < r < m, he eigenfunctions of Th, corresponding to the eigenvalues /i£+r, 0 < r < m, and being orthonornialized with respect to (-y')h- Set Eh = s p a n ( u j , . . . , u j + m ) . Finally, let Ph be the spectral projection of V onto Eh. Similarly as in [5] (Section VIII.5), one has Propos i t ion 4 .1 Let wk € E, then, for sufficiently small h, \\wk-PhWk\\<C.\\(T-th)ivk\\. Corollary 4 .1 6(E,Eh) = sup{d(wk,Eh);wk e E,\\wk\\ =- 1} < C.||Tr - 7*| | . Consequently, from Theorem 2.1, the distance between the two 'eigenspaces' tends to zero with h. Moreover one has Theorem 4.1 There exists a set of eigenfunctions Uk+r, 0 < r < m ofT, corresponding to ftk and being orthonormalized with respect to (•, •), such that, with uh+r, 0 < r < m as above, l l ^ + r - ^ + r l l - 0 i f h - > 0 , 0 < r < m . (4.1) 303 Proof. This adapts the two basic ideas of the proof in [2] (Theorem XIIA .5, p. 907-909), but is more involved. First one defines the non-singular square matrix (ft) = ((iri) by m Phuk+r = 5>H.«£+,., 0 < r < m i=0 and one introduces Uk+t = JT((3-l)tl.uk+h 0<t<m. 1=0 Using Proposition 4.1, one may show that ||£lV{-t — uk+t\\ —> 0, 0 < t < m, if h -» 0. From this convergence, (4.1) can be derived by induction, whence Uk+r is generated from Uk+r, 0 < r < m, by the Gram-Schmidt orthonormalization procedure. I 5 Approximat ion of the bilinear form. The analysis above may be adapted to the case that a(-,-) is suitably approximated on 14 x V̂  and (•, •) is retained exactly. By superposition one arrives at the case where both a(-, •) and (•, •) are suitably approximated on Vh x Vh. Thus define fh : H -> Vh, V/ e H, .Vv <E Vh : ah(fhf, vh) = ( / , vh) , where ah(-,-) is a symmetric, uniformly bounded and uniformly strongly coercive bilinear form on Vh x Vh (uniformly with respect to h), fulfilling a hypothesis similar to (H?). To show the convergence for h —* 0 of the eigenvalues and eigenfunctions of Th, which is a compact, self-adjoint, positive definite operator in H, one proves that |T - Th\ = sup{|(T - fh)v\; v <E H, \v\ < 1} -> 0 if h - 0. R e f e r e n c e s . 1. Chatelin, F. Spectral approximations of linear operators, Academic Press, N. Y. (1983). 2. Dautray, R. and Lions, J-L. Analyse mathematique et calcul nuinerique pour les sciences et les techniques, tome 2, Masson, Paris (1985). 3. Fix, G. Effects of quadrature errors in finite element approximation of steady state, eigen-values and parabolic problems, in Aziz, A. K. (editor), The mathematical foundation of the finite element method with applications to PDE's, Academic Press, N. Y. (1972), pp . 525-550. 4. Kato, T. Perturbation theory for linear operators, Springer-Verlag, Berlin (1976). 5. Mercier, B. Topics in finite-element solutions of elliptic problems, Springer-Verlag, Berlin (1979). 304 

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On an internal approximation of a class of elliptic eigenvalue problems

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Institute of Mathematics AS CR, v. v. i.