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

— 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 developped 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]): PI : l im|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 l) Département de Mathématiques, École Polytechnique fédérale de Lausanne, Suisse. () Department of Mathematics, American University of Beirut, Liban. R.A.I.R.O. Analyse numérique/Numerical Analysis, vol. 12, n° 2,1978 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 ,VüeI f 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 ON SPECTRAL APPROXIMATION. II 115 they will satisfy the relations a f_Kx01*)dz and Fj=(2nO" 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-Bk\\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\.