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RAIRO. A NALYSE NUMÉRIQUE

J EAN D ESCLOUX

N ABIL N ASSIF

J ACQUES R APPAZ

On spectral approximation. Part 2. Error estimates for the Galerkin method

RAIRO. Analyse numérique, tome 12, no2 (1978), p. 113-119

<http://www.numdam.org/item?id=M2AN_1978__12_2_113_0>

© AFCET, 1978, tous droits réservés.

L’accès aux archives de la revue « RAIRO. Analyse numérique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/

conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

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(vol. 12, n° 2, 1978, p. 113 à 119).

ON SPECTRAL APPROXIMATION PART 2. ERROR ESTIMATES FOR THE GALERKIN METHOD (*)

by Jean DESCLOUX (*), Nabil NASSIF (2) and Jacques RÂPPAZ (*) Communiqué par P.-A. RAVIART

Abstract. — One considers an isolated eigenvalue A of finite multiplicity of an operator A which is approximated by a Galerkin method. Using Osborn's technics, one dérives several error estimâtes for X.

1. SITUATION AND RESULTS

In part 1 of this paper [3], we have been concernée! with the problem of convergence in spectral approximation; since the theory we have deve- lopped has received concrete applications for non compact operators only in connection with the Galerkin method, we shall now restrict ourself to this case.

Let X be a complex Banach space of norm 11 11 and { Xh } be a séquence of finite dimensional subspaces of X. One gives two continuous sesquilinear forms a and b on X and one supposes a coercive. Then, by Lax-Milgram, one can define the continuous operators A :X^>X and Ah : Xh—*Xh by

a(Au, v) = b(u, v)> Vw, i?eX? a{Ahuy v) = b(u, v), V

All along this paper we shall suppose that the two following conditions are satisfied (see [3]):

P I : l i m | j ( ^ - ^ ) l^ll =0; P2: VxeZ, lim inf ||x-xA|| = 0.

h-*O h^O xheXh

Let X e C be an isolated eigenvalue of A of finite algebraic multiplicity m ; since a is coercive X ^ 0 and there exists a closed dise À of center X and boundary F such that 0 £ À and A na (A) = {X} where a {A) dénotes the spectrum of A. Let \ilh, . . . , \^m^fh be the eigenvalues of Ah, repeated following their algebraic multiplicities and contained in À. In [3], section 2,

(*) Manuscrit reçu le 10 juin 1977V

l1) Département de Mathématiques, École Polytechnique fédérale de Lausanne, Suisse.

(2) Department of Mathematics, American University of Beirut, Liban.

R.A.I.R.O. Analyse numérique/Numerical Analysis, vol. 12, n° 2,1978

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114 J. DESCLOUX, N. NASSIF, J. RAPPAZ we have proved:

a) m(h) =m for h small enough;

b) lim \xih = X, i = 1, 2, . . . , m.

fc->0

The purpose of this part 2 of our paper is to give estimâtes of X by the

\iih'$. In fact, we shall adapt to the situation described above Osborn's method [ 5 ] ; note that, independently of the fact that Ah is a Galerkin approximation, we have simplified the présentation of Osborn's main argument and strengthened his results. See also Grigorieff [4],

At this point, we recall some standard notations. For an operator £>, Rz (D) = (z—Z))"1 is the résolvent operator. Let Y and Z be closed subspaces of X; then for xe X9

d(x,Z)= inf\\x-z\\, StZZ)^ sup S(y, Z)

zeZ yeY

and

) = max(5(Y,Z),5(Z, Y)).

Let us also open a short parenthesis on duality. Let J5f* be the adjoint space of X, i. e. the set of antilinear continuous forms on X. By Lax-Milgram, the operator C : X * —• X defined by the relation a (v, C cp) = q> (u), V v e X, ( p e l * , is an isomorphism between X* and X which allows to identify these two spaces. With this identification if Z) : X—>X is a bounded linear operator, its adjoint D* :X^>X will be characterized by the relation a (Du, v) = a (w, D* v), V w, v e X; one vérifies also immediately the relation

II^NIKIMIc-MUMI.

We need, for the following, to introducé some further operators. Uh : X—> X is the projector with range Xh defined by the relation a(Hhu — M, V) = 0, V Ü G 4 One has Ah = UkA \Xh and we set Bh = TLhATLh : X~±X; exept for zero, Bh has the same spectrum as Ah and the same corresponding invariant subspaces. E = (2 II i )~M Rz(A)dz is the spectral projector of ^4 relative

J r f

to X and, for /* small enough, Fh = (2 II i ) M J?z (£,,) Jz is the spectral projector of Bh relative to \ilh, . . . , )imh. Now consider the adjoints of these operators as defined above. A* has the isolated eigenvalue X of algebraic multiplicity m; II* will be the projector with range Xh satisfying the relation a{v, H*u — u) = 0 , V ü e If t; E* and F£ will be the spectral projectors of A* and B* — II* A* II* associated respectively to X and to the set \ilhi . . . , \imh; R.A.Ï.R.O. Analyse numérique/Numerical Analysis

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they will satisfy the relations

a f_Kx01*)dz and F j = ( 2 n O "1 \_R2(Bl)dz

where Y is the conjugate circle of Y (positively oriented).

In applications E{X)%aà E* (X), the m-dimensional invariant subspaces of A and A* corresponding to X and X9 will be often composed of smooth functions so that it is reasonable to introducé the quantities

Yft = Ô(£(X), Xh\ y* = 8 (£*(*), Xh).

We can now state the results.

THEOREM 1 : There exists a constant c, independent of h such that î \ E(X)) té cyh; S(F*(X), £*(*)) g cy*.

In section 2, we shall show that Fh \EW defînes for h small enough, a bijection between E(X) and Fh(X); let Ah be this bijection; A = A \E(X)

and Bh = A'1 BhAh will be considered as operators in E(X); A has the eigenvalue X of algebraic multiplicity m and Bh has the eigenvalues

THEOREM 2 : TTzere exw/^ a constant c, independent of h such that

By the choice of a basis in E(X), theorem 2 reduces our original task to a pure matricial problem. Let ƒ be a holomorphic function defined in a neighborhood of X; writting/(^) a n d / ( ^ ) by the mean of Dunford intégrais, one vérifies immediately that

\\ftf)-f(Bà\\

Em

<c\\Â-B

k

\\

Em

where c dépends on ƒ but not on h; using the classical properties of traces and déterminants, one obtains theorem 3 a, b; theorem 3 c, d is a direct application of results quoted in [7], pp. 80-81; hère a is the ascent of the eigenvalue X of A, p is the number of Jordan blocs of the canonical

y\.

form of A.

vol. 12, n° 2,1978

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116 J. DESCLOUX, N. NASSIF, J. RAPPAZ

THEOREM 3 : There exists a constant c, independent of h such that for h smal! enough:

<0 max | X - ^ | g

i = l . . . m

d) min \X-iiih\^c(yhytrm.

i~l...m

REMARKS : 1) In his original work, in a différence context, Osborn [5]

has obtained theorem 3 a for ƒ (z) = z and theorem 3 c; in another context also Chatelin [1] proves, theorem 3 a for f(z) = z.

2) For ƒ (z) = 1/z, theorem 3 a gives an estimate of l/X by the arithmetic mean of the l/\iih's; the result has been already obtained by [2]; we are indebted to Chatelin who showed us that it can also be deduced by Osborn's method.

In order to illustrate this theorem, we consider the example developped in section 4 of part 1 of this paper [3] ; one can prove by Rappaz' method of élimination used in [6] the existence of an infinité number of isolated eigenvalues of finite multiplicities; by supposing the coefficients a, P, . . . sufficiently smooth, one vérifies that the corresponding eigensubspaces are subsets of H 2 x (H1)2; consequently yh = O (h), y* = O (h) and the estimâtes of theorem 3 a, b are of order h2.

We conclude this section by stating a very elementary result for the self- adjoint case. We suppose that the forms a and b are hermitian. Because of its coercivity, a is a scalar product for which X is a Hubert space with norm || x ||f = a(x, x); then A9 Bh and Tlh become hermitian. Let v be an eigenvalue of A9 which is not supposed isolated or of finite multiplicity, and G be the corresponding eigensubspace. For the distance S (v, a (Bh)) from v to the spectrum a (Bh) of Bh> one gets the estimate

ô(v9a(Bh))= inf \\(Bh-v)y\\a^ inf

X X

inf ||(i4-v)y||.= inf \\(A-v)(y-x)

yeXh yX

| y | | = i U

R.A.I.R.O. Analyse numérique/Numerical Analysis

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i. e., since the norms ||.|| and ||.[|a are equivalent:

ô(v, o(Bh)) g c inf 8(x, Xh), c independent of h and v. (1)

xeG

REMARKS: 1) We have obtained the estimate (1) without supposing PI or P2.

2) Examples show that it is not possible to replace the right member of (1), by c{ inf

xeG 1*11 = 1 2. PROOFS

In this section we prove theorem 1 and 2. We use the définitions and nota- tions of section 1 and we suppose hypotheses PI and P2. c will dénote a "generic" constant.

We first recall a well-known resuit. Since a is continuous and coercive, the projectors TLh are bounded uniformly with respect to h and there exists a constant c such that [| x—IIh x || :g c S (x, Xh), V x e X; the n^'s possess the same properties.

Lemma 1 of section 2 of [3] shows that PI implies the inequality sup || Rz(Ah) x || rg c, V z e F for h small enough, c independent of h.

xeXh

II*II = I

We extend this resuit to Bh and i?*.

LEMMA 1 : There exists h0 > 0 and c such that

||Kz(i?ft)|jgc, h<h0, zeT and

\\Rs(B*)\\£c9 h<h0, zeF.

Proof: Since R~ (B*) = (Rz (5ft))* we need t o prove only the first statement;

since Bh is compact it suffices t o verify that || (z—Bh) x\\ ^ c | [ x | | , V xe X, z e T. Taking in account the fact t h a t 0 £ T one has

We note that we shall not use any more PI explicitely. Consequently, in the proofs of lemma 3 and theorem 1, the statements for the adjoints operators are obtained in the same way as for the direct operators.

We omit the proof of the foliowing trivial :

LEMMA 2 : Let Y and Z be two subspaces ofX with the same finit e dimension;

1 et P: Y-> Zbe a linear operator such that \\ Py-y \\ g 0.5 || y ||, V^ e Y,

vol. 12, n° 2,1978

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1 1 8 J. DESCLOUX, N. NASSIF, J. RAPPAZ

Then P is bijective,

| | P - z | | ^ 2 ! |Z| | , VzeZ and

sup | | p -1z - z | | g 2 sup

zeZ yeY

LEMMA 3:

|j(£-Ff t)

\\(E*-F*)

Proof: For h small enough, by lemma 1, one has

\\(E-F

h

) |

Bm

||^(2nr

1

J

r

||jR,(B

A

)|

^c\\(A-Bh) \Em\\;

\\(A-Bh) \Em\\^\\(I-Uh)A \E(X)\\ + \\nhA(I-nh) \Em\

Proof of theorem 1; Lemma 3 implies that &(E(X), Fh(X)) ^ cyh. Set, as in Section 1, Ah = Fh \E(X) : E(X) ->Fh(X); for h small enough E{X) and Fh (X) have the same dimension m; on the other side P2 implies lim yh = 0;

by lemma 2, A^1 exists for h small enough and is uniformly bounded with respect to h; furthermore sup || A^"1 x — x || ^ c yh9 i. e*

e F C )h C )

Proof of theorem 2: Let iSft = A^1 Fh—I : X—• X; Shis uniformly bounded with respect to h (see proof of theorem 1); from the identity

A-Bh)x, xeE(X), one obtains for xeE(X), y e E* (X), since Fh Sh = 0,

a((A-Bh)x, y) = a((A-Bh)x, y) + a(Sk(A-Bh)x, (I~F*)y); (2) sup a((Â-Bh)x, y)

x<=E(X),yeX ll*ll = l l y | | = i

sup a((A-BJx9y); (3)

xeE(X),yeE*(X) 11*11-11*11 = 1

R.A.I.R.O. Analyse numérique/Numerical Analysis

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for xeE(X), yeE*(X), || x |[ = || y || = 1, one has (using lemma 3):

laiS^A-BOx^I-F^yy^cWiA-B^xW.Wil-F^yW^cy.yt; (4) a((A-Bh)x, y) - a ((1-1104 *, (I-U*)y)

+ a((I-nh)x,(I-Il*)A*y)

\ \ (5)

theorem 2 follows from (2), (3), (4) and (5). •

REFERENCES

1. F. CHATELIN, Théorie de Vapproximation des opérateurs linéaires, application au calcul des valeurs propres d'opérateurs différentiels et intégraux. Notes de cours, Université de Grenoble, 1977.

2. J. DESCLOUX, N. NASSIF and J. RAPPAZ, Spectral Approximations with Error Bounds for Non-Compact Operators, Rapport du département de mathéma- tiques E.P.F.L., 1977.

3. J. DESCLOUX, N. NASSIF and J. RAPPAZ, On Spectral Approximation, Part 1:

The Problem o f convergence', this journal.

4. R. D. GRIGORIEFF, Diskrete Approximation von Eigenwertproblem, Numerische Mathematik; part I: Vol. 24, 1975, pp. 355-374; part I I : Vol. 24, 1975, pp. 415-433; part III : Vol. 25, 1975, pp. 79-97.

5. J. E. OSBORN, Spectral Approximation for Compact Operators, Math. Comp., Vol. 29, 1975, pp. 712-725.

6. J. RAPPAZ, Approximation of the Spectrum of a Non-Compact Operator Given by the Magnetohydrodynamic Stability of a plasma, Numer. Math., Vol. 28, 1977, pp. 15-24.

7. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, 1965.

vol. 12, n° 2, 1978

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