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A two-grid discretization scheme for eigenvalue problems. (English) Zbl 0959.65119

A two-grid discretization technique is presented for solving linear eigenvalue problems posed in variational form,


Here, X denotes a real Hilbert space, and a(,) and b(,) are symmetric and positive bilinear forms such that the problem exhibits a point spectrum 0<λ 1 λ 2 . Two linear finite element subspaces S H S h X are introduced which are defined on a coarse grid T H and a refined grid T h , respectively. Given on the coarse grid an approximate eigensolution u H S H , λ H R 1 , then u h S h , λR 1 is defined by a(u h ,v)=λ H b(u H ,v) for all vs h together with λ h =a(u h ,u h )/b(u h ,u h ). Asymptotic error estimates in terms of H and h are presented. The technique is illustrated by simple numerical experiments for the first Laplace eigenvalue problem on the unit square.

65N25Numerical methods for eigenvalue problems (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
65N15Error bounds (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35P15Estimation of eigenvalues and upper and lower bounds for PD operators