RAIRO ANALYSE NUMÉRIQUE

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1 RAIRO ANALYSE NUMÉRIQUE TERESA REGIŃSKA External approximation of eigenvalue problems in Banach spaces RAIRO Analyse numérique, tome 18, n o 2 (1984), p < 18_2_161_0> AFCET, 1984, tous droits réservés. L accès aux archives de la revue «RAIRO Analyse numérique» implique l accord avec les conditions générales d utilisation ( legal.php). Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

2 R.A.LR.O. Analyse numérique/numerical Analysis (vol. 18, n 2, 1984, p. 161 à 174) EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS IN BANACH SPACES (*) by Teresa REGINSKA (*) Communicated by Fr. CHATELIN Abstract. We are concerned with approximate methods for solving the eigenvalue problem Tu Xu, u ^ 0, for the linear bounded operator T in a Banach space X. The problem is approximated by an appropriate family of eigenvalue problems for operators { T h }. We present a theoretical framework which allows us to consider in the same way the methodsfor which T h are defined on subspaces of X and those which are defined on spaces forming external approximation of X. Particularly, the paper contains theorems on sufficient conditions for stability andstrong stability of{ T h }. Résumé. On considère ici une classe de méthodes de résolution approchée du problème spectral de la forme Tu = Xu, où T est un opérateur linéaire, borné dans un espace Banach X. Les méthodes présentées remplacent le problème original par une famille de problèmes spectraux pour des opérateurs T h. Les résultats sont présentés d'une manière qui permet de considérer à la fois les méthodes où les T h sont définis sur des sous-espaces de X et celles où les espaces de définition de T h forment une approximation externe de X. Vouvrage contient certaines conditions suffisantes de stabilité et de stabilité forte de la famille {T h }. 1. INTRODUCTION Let X be a Banach space and T e 5 {X) be a linear bounded operator on X. Let us consider the eigenvalue problem Tu = Xu, u # 0. Most methods used to solve this problem consist in approximation of the initial problem by a séquence of eigenvalue problems for T h e <$?(X h \ where X h are fini te dimensional subspaces of X and T h are certain approximantes of T. This approach has been used in many papers, among others by J. Decloux, N. Nassif, J. Rappaz in [5] and by F. Chatelin in [2], However, there are methods which cannot be presented within this unifying theoretical framework (e.g. the Aronszajn's method, cf. [1, 12]). Therefore we consider the more gênerai case of approximation when the operators T h are defined in spaces not contained in X. Strictly speaking we use an external approximation of X. We present some theorems (*) Received in October 1982, revised in May O Institute of Mathematics Polish Academy of Sciences, Sniadeckich 8, skr. poczt Warszawa, Poland R.A.LR.O. Analyse numérique/numerical Analysis, /84/02/161/14/$ 3,40 AFCET Bordas-Dunod

3 162 T. REGINSKA concerning the approximation of eigenelements of T by eigenelements of T h. Particularly we formulate new theorems about sufficient conditions for stability and strong stability of { T h }. Let us introducé a family of Banach spaces { X h } hejf with the norms A, where ffl a U + has an accumulation point at 0. We assume that there exist uniformly bounded linear maps r h : X on > X h. Let F be a normed space such that there exist an isomorphismcû : X - F and uniformly bounded linear maps p h : X h -> F. We adopt the following définition : DÉFINITION 1 : An approximation { X h, r h, p h } of X is said to be an external approximation convergent in F if for any u G X lim (ÙU - p h r h u \\ F = 0. The above définition is weaker than that used customarily (cf. [11, 6]). Next, let us introducé a family { T h } he^ of linear operators where T h e ^(X h ), We will assume that : Al : The approximation { X h, r hi p h } of X is convergent in F; A2 : For any u e X lim r h Tu - T h r h u \\ h = 0. ho 2. STABILITY OF { T h } Let p(t) and p(t h ) dénote, as usually, the résolvent sets of operators T and T h respectively. We additionally assume that either the operators T h have no residual spectrum or that the residual spectrum of T h does not contain the points of p(t) (since not only fmite dimensional approximation is considered). We will use the following définition of stability cf. [4, 2] : DÉFINITION 2 : The approximation {T h } is stable at z e p(t) iff3h(z), Mh < h(z) : z e p(t h ) and (z - T^" 1 ^ M{z) < oo. Now we are going to formulate some sufficient conditions for stability of { T h } in tenus of external approximation of T. Let N(r h ) dénote the null space of r h. Let us introducé the set of families of complementary subspaces of N(r h ) in X ^ = { { y H }*«*, V h ax,v h N(r h ) = X}. LEMMA 1 : If there exists { V h } h e^ e SF such that vevh \\v\\ = l PhT h r h v\\ F^0, (2.1) R.A.LR.O. Analyse numérique/numérical Analysis

4 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS 163 e = e(v h ) == sup mt> -p h r h v\\ F^0, (2.2) vev h \\v\\ = l then { T h } is stable at any X e p(t). Proof: Let X e p(t). Hence, there exists c > 0 such that (?, - 70 u > c u Vwel, and for c = c/\\ CÖ" 1, (Ù(X - T) u F ^ c \\ u \\ Vw e X. Let us take an arbitrary u h e X h, Then there exists v h e V h such that r h v h = u h. We have I v h > (l/d) w h fc and Vx h e X ft xj h > l/d \\ p h x h F, where for any A. Hence (X - T h ) u h \\ h =\\(X- T h ) r h v h \\ h ^ 2 = \\\<ù(x-t)v h + X( Ph r h - o) v h d x ^ i II u h \\ h (c~\x\^h ~5 h ). Thus,forgivenXG p(t) there exists h 0 such that for h < h 0 what means, according to Définition 2, that { T h } is stable at X. Remark 1 : In the case of an internai approximation of X, when F = X, X h = V h cz X and œ and p h are identity maps, and r h are projections of X on X h, the condition (2.2) is automatically satisfied with s h = 0. In turn, the condition (2.1) takes the form (T T h ) \ X h ^0 i.e. the assumption of Lemma 1 in [5]. In the gênerai case of an external approximation we have E h # 0. Thus, we must analyse how s(v h ) dépends on { V h } e êf. To do this we introducé the following numbers characterizing the subspaces V h : y (F,)- sup Ö* Hl, (2.3) vev h \\v[\ = l where Q h (h e Jf ) are some given linear and bounded projections of X onto N(r h ). Let V h = (1 - Q h ) X. In this case y(v h ) = 0. VOL. 18, N 2, 1984

5 164 T. REGINSKA We can state the following resuit : LEMMA 2 : Let us assume that z(v h ) - 0 as h -> 0. Then e(v h ) - 0 for {V h }e^ if and only ify(v h ) -> 0. Proof : e(v h ) = sup <ùq h v + (1 - GO t> - />* r fc (l - Q h )v \\ F > uek h 11*11=1 -(1 +y(v h ))e(v h ). The implication " => " follows from the above inequality. It is easy to see that s(v h ) < sup { œ. Q h v + (1-00 v 8(F ft )} < y(v h ) \\ œ lkll = i which ends the proof of Lemma 2. In the case when the X h are infinité dimensional spaces the condition (2.2) becomes very strong, so another version of Lemma 1 will be more useful in this special case. Let us introducé the following DÉFINITION 3 : The family {V h }, V h cz X is asymptotically equivalent to { X h } with respect to {r h } {r h G JS? (X, X h \ r H X = r h V h = X h ) if the r h are uniformly boundedand inf r h x \\ h ^ c > 0, VA G 3%. xev h LEMMA 3 : Ifthere exist {r h } and { V h } asymptotically equivalent to { X h } with respect to {r h } such that ê(v h )-.= sup KT-CrJ^-^rOHI-O, vevh t; =1 then {T h } is stable at any X e p(t). Proof : Let us take u h e X h. Let v h G V h be such that r h v h = u h : II (>- ~ T h )u h \\ h = (X - TO?* «fc L =? (? l^)" 1 (A. - T h ) r h v h \\ h ^ >c\\xv h - Tv h +(T-(r h \ Vh y 1 T h r h )v h \\ >c\\(x-t)v h \\- $(V h ) v h. R.A.I.R.O. Analyse numérique/numerical Analysis

6 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS 165 Smeek G p(t), there exists a constant c 1 > Osuchthat (X T)v h \\ ^ c 1 v h \\. Moreover, v h ^ ^ r I r \\ u h \\ h. Iic 2 ' = sup r h, then h what proves Lemma 3. Now, we are going to give a short analysis of the assumptions of the above lemma To do this we restrict our considérations to the case of separable Hilbert spaces. LEMMA 4 : For an arbitrary separable Hilbert space X andafamily of separable Hilbert spaces X h there exist uniformly bounded maps r h : X - X h such that the orthogonal compléments of the null spaces of r h form a family asymptoticaïly equivalent to { X h } with respect to { r h }. Proof: Let { u } =i and { u h n } =1 be orthonormal bases in X and X h respectively. If X h is /c-dimensional, we put wj = 0 for j > k. Transformation cp : X - l 2 and cp ft : X h - l 2 are defined as follows : <pu - { (M, U X \ (M, M 2 ),... } for wel, <p fc v = { (Ï? S MÏ) fcï (i>, «5) fc,... } for i)el ft. Thus Vu G X cpw j l2 = M and V { x n } G l 2 Similarly cp ft = 1 and cp^" 1 : cp fe X h -+ X h, <p~ x = 1. Let P h be the orthogonal projection from l 2 onto cp,, X h. Let T ft :== Cp j r h Cp. A > A^, ^Z. DJ For any ve X \\ r h v \\ h ^ \\ v \\ and since (pv h = cp ft X h9 r h \ Vh = cp^" 1 9 \ Vh and (r h I^J" 1 = cp" 1 cp ft. Thus (r h Fh ) -1 = 1. Hence { V h } is asymptotically equivalent to { X h } with respect to { r h }. Now, let us take arbitrary éléments v ev h and x G N(r h ). For v there exists u v e X h such that (1;, M ) = (w y, MJ), / = 1, 2,... Hence (v, x) - ^ K> w?) ( x ' w i)- VOL. 18,N 2, 1984

7 166 T. REGINSKA Since cpx _L <p h X h9 O, w.) (u, u\) = 0 for any u e X h9 so it also holds for u = u v. Thus (v, x) i = 1 0 for any ve V h and x e N(r fc ), what means that V h is orthogonal to N(r h ). Let Q ft be orthogonal projection onto N(r h ), and V h be complementary subspace of N(r h ) in X. Thus inf r h v \\ h - inf I r h Q h v + r fc (l - ÔJ ^ I = vevh \\\ vevh = inf (l-ôfch Iio-e*) H inf Cl O*) «- inf \\r h x\\ h. vevh \v\\ = l x±n(rh) x -l Using the notation (2.3) we obtain inf IMI=1 I r h v inf This leads us to the following remark ; Remark 2 : Let { iv^) 1 } be asymptotically equivalent to { X h } with respect to { r h }. If 3c 0 > 0 such that V/z < /* y(t h ) ^ c 0 then the family { V h } is also asymptotically equivalent to { X h } with respect to { r h }. Remark 3 : If { F h } satisfies the condition (2.2), then { tically equivalent to { X h } with respect to { r h }. This follows from the inequalities : Vu e F h, Ü = 1 : } is asympto- tg il A ^~ l II Since p h \\ ^ a and coi; CÛ u, we have ^ - for any v e V h and i; = 1. R.A.I.R.O. Analyse numérique/numerical Analysis

8 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS APPROXIMATION OF EIGENELEMENTS OF T In this section the proofs of the theorems are based on the ideas contained in [5] and [2]. Let F be a Jordan curve in the résolvent set p(t). If { T h } is stable for all XeT, then V c p(t h ) for sufficiently small h < h 0. Hence, we can defïne the spectral projectors E : X -» X and E h : X h - X h by LEMMA 5 : If the assumption A2 is satisfied and {T h } is stable on F, then \fv G X lim r h Ev - E h r h v \\ h = 0. h-o Proof : From the définition of E and E h and from the identity r h (X - T)-> ~(X- TJ- 1 r h = (X - T h YHT h r h - r h T)(X - T)~ x it follows that for given ve X \\r h Ev - E hh v\\ <JIisup fl, - T h y\t h r h - r T)(X - TT 1 Hl = where U = {we X :u = (X - T^-^^eT}. The operators (X T^1 are uniformly bounded for X e F and h < h 0 since the stability of { T h } on F. Thus, by the assumption A2, Moreover, V«e A" (X - T h y \T h r h - r h T) u -» 0. (X-TO-^r^-r, T) < (X-TJ" 1 r T + X(X- T,)" 1 r, + r h, so the operators (X T h )~ 1 (T h r h r h T) are uniformly bounded for and h < h 0. Thus, since the set U is compact, sup I (X - T h y \T h r h -r h T)u ->0. VOL. 18, N«

9 168 T. REGINSKA LEMMA 6 : If Al and A2 are satisfied and {T h } is stable on F, then Vu e EX inf cou p h y h \\ F -> 0. Proof: Since inf \\(ùv-p h y h \\ F ^ \\<ov-p k r h v\\ F + \\p h \\ \\ r h Ev - E h r h v, the proof follows immediately from Lemma 5. As usually, <J(T) dénotes the spectrum of T. Let Q c C be an open domain with the boundary F c= p(t) which is a Jordan curve. Finally, let THEOREM 1 : If the assumptions A1 and A2 are satisfied and {T h } is stable in p(t) then : 1 if Q n o(t) ^ 0 then o(t h ) n Q / 0 for sufficiently small h, 2 if k 0 G a(t) and 3Ô O > 0 : K(k 0, 5 0 ) n a(t) = { X o } then VO < 5 < 8 0, 0 ^ <j(t h ) n X(A, 0, 5 0 ) c= K(k 0, Ö) /or sufficiently small h, 3!ƒ ^ G o(t h ) and X h -+ X o then X o G a(t). Proof: It follows from Lemma 5 that VÜ e EX inf \\r h v y h \\ h - 0. If u # 0 then, since Al, r h v / 0 for sufficiently small /*. Thus 1 is proved. For the proof of 2 it is enough to remark, that for 0 < S < 8 0 K(X, 6 0 )\int K(A, 5) c p(t) and thus, by the stability of { T h }, K(k, ô o )\int K(k, S) is contained in p(t h ) for /i < h 0. Assume now that X h G o(t h ) and A ḣ -> X o ^ a(t). Thus there exists S > 0 such that K(A, 0, S) c p(t) and from the stability K(^o, S) c p(t h ) for h < h 0, what means that for h < A ls X ft G p(t h ). The above theorem gives convergence of eigenvalues, but without préservation of the algebraic multiplicities. Namely, we have only THEOREM 2 : If A1 and A2 are satisfied and {T h } is stable on F then 1 dim EX = oo => dim E h X h -> oo 2 dim EX = n ^ dimp h E h X h ^ n. : Let { u t } = x be a linearly independent set of éléments of X From Lemma 6 it follows that for every finite number N Ve 3h e VA < h e Vi - 1,...,AT 3x? e E h X h : cm/; - /? h JC* F < e. R.Â.I.R.O. Analyse numérique/numerical Analysis

10 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS 169 Thus ViV < oo 3h N VA < h N dimp h E h X h ^ JV, hence 1. Let now dim EX = n. By Lemma 6 we have sup inf ou - p h y h \\ p -> 0. ve X y he hx ll"ll = l Using the known notation (cf. [7] chap. IV) : for closed subspaces 7, Z of X 8(7, Z) = sup inf >> -z, (3.1) we have 5(<tiEX,p h E h X h ) - 0. It is known that if 8(7, Z) < 1 then dim 7 ^ dimz(c/ [7] chap. IV, Corollary 2.6). Thus n = dim coex ^ dim p h E h X h. Under additional assumptions we can state the following result : THEOREM 3 : One supposes Al, A2 and stability of {T h } on F. Moreover let p h u h f \\ F -> 0, where u h e X h, imply that f belongs to cox, and let the norms in F and X h be asymptotically equivalent (Le. ifu h e X h and \\ p h u h \\ F -> 0 then u h \\ h^> 0). Then if x h e E h X h and \\ p h x h - f \\ F -> 0 then f e (oex. Proof : If p h x h f - 0 then there exists x o e X such that ƒ = cox 0. It remains to show that Ex 0 = x 0. From the inequality we get kxo\\f-\\phek(r h Xo-x h )\\ P II Ex 0 - x 0 ^ co~ * II [II CÖX 0 - p h x h II F + II mex 0 - p h r h Ex ö \\ F + + lift I \\r h Ex 0 -E h r h x 0 \\ h + \\p k E h \\ lk x 0 - xj ]. The convergence \\p h x h cox o -> 0 implies /? fc r fc x 0 - p h x h F -> 0 and thus, by the additional assumption on p h, \\ r h x 0 x h \\ h - 0. By Lemma 5 and ^41 we have : Ve 3h 0 VA < A o Ex 0 x 0 \\ ^ e, thus ^XQ = x STRONG STABILITY OF { T h } Let D e C be a domain limited by the Jordan curve F ei p(t). Let E and E h be the spectral projections associated with the spectrum of T and T h inside F. We will assume that dim EX < oo. With respect to the convergence of eigenvectors it is very important to have the same dimensions of E h X h (orp h E h X h ) VOL. 18,N<>2, 1984

11 170 T. REGINSKA and EX. We will use the notion of strongly stable approximation { T h } similar to that introduced by F. Chatelin in [4]. DÉFINITION 4 : An approximation { T h }, stable on F, is strongly stable on F if dim EX = dim/?,, E h X h for h small enough. The convergence of external approximation (i.e. A\\ the consistency of { T h } to T (i.e. A2) and the stability of { T h } are not sufficient for strong stability of { T h }, so we need a stronger assumption. LEMMA 7:If{T h } is stable on F and \\(T h r h -r h T)(X-T)-i\\ h^0 for XET (3.2) then \\r h E~ E h r h \\* ixm -> 0. Proof : Repeating argumentation of the proof of Lemma 5 we get r h E - E h r h ^ c 0 (T h r h - r h T)(X - T)' 1 for a some constant c 0. LEMMA 8 : If there exists { V h } e 3F such that V/i < h 0 then i\ h := inf \\p h r h x\\ F ^ s 0 > 0 xev h 11*11 = 1 (Ph E h x h> EX ) < \\Pk E h r h -. e o Proof : Let V h be a subspace of V h such that r ft V h = E h X h. Then II p h E h r h - (ùe ^ sup inf p h E h r h x - coy \\ ^ ^ sup inf p h r h x~ay \\ > inf \\p h r h x\\ sup inf /? h x h -(oy \\. 11*11 = 1 11*11 = 1 l Ph*h -l According to (3.1) the last factor is equal to 6(p h E h X h, coex). THEOREM 4 : Ifthe assumptions A1, (2.1), (2.2), (3.2) are satisfied, then {T h } is strongly stable on F. Proof.lt follows from (2.2) that inf cox F sup p p p h r hh h x F xev h X ev h II 11*11 = 1 11*11 = 1 R.A.I.R.O. Analyse numérique/numencal Analysis i

12 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS 171 thus v[ h ^ e 0 > 0 for sufficiently small h. Moreover, since dim EX < oo, by Lemma 7 \\p k E h r h -<ùe\\ ^ lift \\E h r h -r h E\\ + (p h r h - ) -> 0. Hence, from Lemma 8 we get 8{p h E h X h, (oex) < 1 for h small enough and thus dim p h E h X h < dim G>EX. The oposit inequality have been obtained in Theorem 2, thus dim p h E h X h = dim EX. The assumption (2.2), which is very strong in the case of infinité dimensional spaces X h, can be ommited as it is shown in the following. THEOREM 5 : Let A\ be satisfied. Moreover, let {V h } be asymptotically equivalent to { X h } with respect to { r h } and { X h } be asymptotically equivalent to {p h X h } with respect to {p h }. If \\[T~(r h \ v X 1 T h r h ](X~T)~ l \\^0 for XeT (3.3) then {T h } is strongly stable on T. Proof: It follows from (3.3) that 3c> 0 V/* < h 0 V^ e T \\(r h WX'Q - T h ) r h (X - T)" 1 ^ c. On the other hand Ur h \ v r i^- T k)'-h^-t)- 1 H\\X-T h \\ \\(r h \ Vh )- l \\ I Ml (X-T)- l. Thus, by the uniform boundness of (^l^)' 1 and r h \\ we obtain that X - T h \\ ^ Ci > 0 for h < h 0 and À, e T, what gives the stability of { T h } ont. Moreover, (3.3) implies (3.2). Thus, by Lemma 7, \\r h E- E h r h -> 0, what implies p h E h r h we \\ - 0, since dim EX < oo. The assumption on asymptotic équivalence of { V h }, { X h } and { p h X h } guaranties the existence of positive lower bound for r\ h. Hence, by Lemma 8,?>(p h E h X h, (ÖEX) Thus dim p h E h X h ^ dim (oex what together with Theorem 2 gives : dim p h E h X h = dim E h X h = dim EX for sufficiently small h. The condition (3.3) imposed on the approximation is some modification of radial convergence introduced in [2,3] for the case of internai approximation. VOL. 18, NO 2, 1984

13 172 T. REGINSKA 5. APPLICATION Let X be a Hubert space with the scalar product a(,). Let b be a bounded sesquilinear form defined on X x X. The eigenvalue problem for two forms b(u, v) = \a(u, v) VveX (5.1) is considered This problem is equivalent to the eigenproblem for an operator T defined by : b(u, v) = a(tu, v) Vw, v e X. Let V be a dense subspace of X. We wilî consider approximate methods of solving the problem (5.1) which are generated by séquences of sesquilinear forms a n and b n defined on F x F. It is assumed that a n n = 0, 1,... are symmetrie and positive défini te and b n are bounded with respect to a n, i.e. Vw, V G V \ b n {u, v) ^ c n a* l2 (u, u) al i2 (v, v). Let X n be the closure of V in the norm a\ 12, n = 0, 1,... The n-th approximate eigenvalue problem has the form flnd XeC and 0 # u e X such that (5.2) which is equivalent to the eigenproblem for an operator T n defined by a n and b n : b n (u, v) = a n (T n w, v) Vu e F, w e X n. Under the assumptions a 0 ^ a n ^ a, (5.3) a is quasi-bounded with respect to a 0, i.e. there exists a symmetrie operator L in X o, with dense domain F, such that a(u, v) = a o (Lu, v) Vu,veV (cf. [1]), (5.4) the approximation (5.2) can be described in terms of external approximation (for details see [8]). From (5.3) and (5.4) it follows that a is quasi-bounded with respect to a n, n = 1, 2,... Let L n be the symmetrie operator defined by a(u, v) = a n (L n u, v) Vu, v e F, and let L n dénote its selfadjoint extension in X n. L n is positive definite. Thus, there is a unique positive definite and self-adjoint square root L* /2 of L n and the domain D(L n ) of L n is dense in D(L^/2 ). It can be proved (see [8]) that D(L 1/2 n ) = X and Vw, vsx a(u, v) = a {L^2 u, L 1/2 n v). Let us put r n := L 1/2 n. It is easy to show (see [8]) that r n \\# {X,x n ) ~ I r ü l W&(x n x) = ^- ^ e define p n -= r~ x. The approximation { X n, r n, p n } is convergent in X due to Définition 1. The following property can be proved (see [8]) : R.A.LR.O. Analyse numérique/numerical Analysis

14 EXTERNAL APPROXIMATION OF EIGENVALUE PROBLEMS 173 LEMMA 9 : Let (5.3) and (5A)be satisfied and moreover VueV sup a (w, v) - a(u, v) \ - 0, (5.5) vev \\v\\ = l sup b n (u, v) b(u, v) -> 0. (5.6) Let u n B < Mand w B B < Mn = 0, 1,... for some M. If a n (u n, w) - Û(M, W) VH> e K, and a n (ü ns w) - a(t;, w)vwek imply 6«(«i,^«)-^ 6(ii, Ü), (5.7) then { T n } is stable at any Xep(T). Let us remark, that in the considered case the condition (2.1) of Lemma 1 implies A2 and (3.2). Thus we have COROLLARY 1 \ If the assumptions (5.3)-(5.7) are satisfied then the method is convergent in the sense of Theorems 1 to 4. The class of methods described above has been investigated by R. D. Brown in [1] by using the another theory. He adopts the theory of discrete convergence of Banach spaces in the form developed by Stummel [10]. His results are similar to those obtained above. REFERENCES 1. R. D. BROWN, Convergence of approximation methods for eigenvalues of completely continuons quadratic farms, Rocky Mt. J. of Math. 10, No. 1, 1980, pp F. CHATELIN, The spectral approximation of linear operators wit h applications to the computation of eigenelements of differential and intégral operators, SI AM Review, 23 No. 4, 1981, pp F. CHATELIN, J. LEMORDANT, Error bounds in the approximation of eigenvalues of differential and intégral operators, J. Math. Anal. Appl. 62, No. 2, 1978, pp F. CHATELIN, Convergence of approximation methods to compute eigenelements of linear operators, SIAM J. Numer. Anal. 10, No. 5, 1973, pp J. DESCLOUX, N. NASSIF, J. RAPPAZ, On spectral approximation, Part 1 : The problem of convergence, Part 2 : Error estimâtes for the Galerkin method, RAIRO Anal. Numer. 12, 1978, pp R. GLOWINSKI, J. L. LIONS, R. TRÉMOLIÈRES, Numerical analysis of variational inequalities, T. KATO, Perturbation theory for linear operators, Springer Verlag, Berlin, T. REGINSKA, Convergence of approximation methods for eigenvalue problem s for two farms, to appear. VOL. 18, N 2, 1984

15 174 T. REGINSKA 9. T. REGINSKA, Eigenvalue approximation, Computationaï Mathematics, Banach Center Publications. 10. F. STUMMEL, Diskrete Konvergenz linearer operatoren, I Math. Ann. 190, 1970, 45-92; II Math, Z. 120, 1971, pp R. TEMAM, Numerical analysis, H. F. WEINBERGER, Variational methods for eigenvalue approximation, Reg. Conf. Series in Appl. Math. 15, R.A.I.R.O. Analyse numérique/numerical Analysis