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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,

$a\left(u,v\right)=\lambda b\left(u,v\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}v\in X·$

Here, $X$ denotes a real Hilbert space, and $a\left(,\right)$ and $b\left(,\right)$ are symmetric and positive bilinear forms such that the problem exhibits a point spectrum $0<{\lambda }_{1}\le {\lambda }_{2}\le \cdots$. 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 $a\left({u}^{h},v\right)={\lambda }^{H}b\left({u}^{H},v\right)$ for all $v\in {s}^{h}$ together with ${\lambda }^{h}=a\left({u}^{h},{u}^{h}\right)/b\left({u}^{h},{u}^{h}\right)$. 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.

MSC:
 65N25 Numerical methods for eigenvalue problems (BVP of PDE) 65N55 Multigrid methods; domain decomposition (BVP of PDE) 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 35P15 Estimation of eigenvalues and upper and lower bounds for PD operators