Eigenvalue approximations by mixed methods

— We study an abstract eigenvalue problem related to a saddle-point formulation, and then we discretize it, obtaining error bounds for eigenvalues and eigenfunctions. We apply our results to the solution of four th order problems by mixed finite element methods. INTRODUCTION We are interested in the following eigenvalue problem: Find X e R and (u, >\f) e V x W such that v)+b(v9 v|/) = 0, VweF, ( (where V, W, H are real Hilbert spaces with W Ç H, a (u, v) and b (v, cp) are continuous bilinear forms on VxV and VxW respectively). Such a problem suggests an abstract scheme for numerical approximation of spectral boundary-value problems for elliptic operators, by means of finite éléments of mixed type. We recall that in the last years the use of mixed methods for solving steadystate problems has been studied by a large number of authors : here, we only meütïön the ateixact-works by^Babuska f t ] afid-Brezzi p ] * 4he—papers ̂ by Babuska-Oden-Lee [3], Raviart-Thomas [17], Thomas [18] for Second Order Problems, and by Brezzi-Raviart [8], Ciariet-Raviart [9], Giowinski [11] for Fourth Order Problems; further références can be found in these papers. For the finite element approximation of eigenvalue problems in the compact selfadjoint case, the main références are the paper by Birkhoff-de BoorSwartz-WendrofT [5] and the book by Stang-Fix [19], whose ideas have largely inspired our work; for the nonselfadjoint case we mention the works by Babuska-Aziz [2], Bramble-Osborn [6], Fix [10] and Osborn [16]. (*) Reçu le 5 août 1977. () This work was partially supported by the C.N.R.-G.N.A.F.A., and suggested by Prof. F. Brezzi during a series of seminars held at the Istituto Matematico del Politecnico, Turin, in December 1976. () Istituto Matematico del Politecnico di Torino. R.A.LR.O. Analyse numérique/Numerical Analysis, vol. 12, n° 1, 1978