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, \[ a(u,v)= \lambda b(u,v)\quad\text{for all }v\in X. \] 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 \(0<\lambda_1\leq \lambda_2\leq\cdots\). Two linear finite element subspaces \(S^H\subset S^h\subset X\) 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 \(u^H\in S^H\), \(\lambda^H\in R^1\), then \(u^h\in S^h\), \(\lambda\in R^1\) is defined by \(a(u^h, v)= \lambda^Hb(u^H, v)\) for all \(v\in s^h\) together with \(\lambda^h= a(u^h, u^h)/b(u^h, u^h)\). 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.


65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
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