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Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems. (English) Zbl 1236.65143

The authors consider the symmetric elliptic variational eigenvalue problem

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

where

$a\left(u,v\right):={\int }_{{\Omega }}\left(\sum _{i,j=1}^{d}{a}_{i,j}{\partial }_{i}u{\partial }_{j}v+cuv\right),\phantom{\rule{1.em}{0ex}}b\left(u,v\right):={\int }_{{\Omega }}\beta uv$

and $V:={H}_{0}^{1}\left({\Omega }\right)$ or $V:={H}^{1}\left({\Omega }\right)$. The problem is discretized using conforming and nonconforming finite elements.

The two-grid discretization scheme is given by

$\begin{array}{cc}\hfill \text{Step}\phantom{\rule{4.pt}{0ex}}\text{1.}\phantom{\rule{2.em}{0ex}}& {a}_{H}\left(u,v\right)={\lambda }_{k,H}b\left({u}_{k,H},v\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}v\in {V}_{H},\hfill \\ \hfill \text{Step}\phantom{\rule{4.pt}{0ex}}\text{2.}\phantom{\rule{2.em}{0ex}}& {a}_{h}\left(u,v\right)={\lambda }_{k,H}b\left({u}_{k,H},v\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}v\in {V}_{h},\hfill \end{array}$

where the indices $H$ and $h$ refer to a coarse grid ${\pi }_{H}$ and a fine grid ${\pi }_{h}$, respectively, and the bilinear form ${a}_{h}$ denotes the elementwise evaluated bilinear form $a$.

Based on results from the abstract discretization theory of eigenvalue problems the authors prove error estimates corresponding to the approximation properties of the trial space. Numerical examples are provided for ${\Omega }$ being a square or an $L$-shaped region.

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