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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(u,v)=λb(u,v)forvV,

where

a(u,v):= Ω i,j=1 d a i,j i u j v + c u v,b(u,v):= Ω βuv

and V:=H 0 1 (Ω) or V:=H 1 (Ω). The problem is discretized using conforming and nonconforming finite elements.

The two-grid discretization scheme is given by

Step1.a H (u,v)=λ k,H b(u k,H ,v)forvV H ,Step2.a h (u,v)=λ k,H b(u k,H ,v)forvV h ,

where the indices H and h refer to a coarse grid π H and a fine grid π 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 Ω being a square or an L-shaped region.


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