## 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 $a(u,v)=\lambda b(u,v) \text{ for } v \in V,$ where $a(u,v):= \int_\Omega \Bigg( \sum_{i,j=1}^d a_{i,j} \partial_i u \partial_j v +cuv \Bigg), \quad b(u,v):= \int_\Omega \beta uv$ and $$V:=H_0^1(\Omega)$$ or $$V:=H^1(\Omega)$$. The problem is discretized using conforming and nonconforming finite elements.
The two-grid discretization scheme is given by \begin{aligned} \text{Step 1.} \qquad &a_H(u,v)=\lambda_{k,H} b(u_{k,H},v) \text{ for } v \in V_H, \\ \text{Step 2.} \qquad &a_h(u,v)=\lambda_{k,H} b(u_{k,H},v) \text{ for } v \in V_h, \end{aligned} 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 for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs
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