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

Yang, Yidu; Bi, Hai

SIAM J. Numer. Anal. 49, No. 4, 1602-1624 (2011).

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.

Reviewer: Rolf Dieter Grigorieff (Berlin)

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

Keywords:

second order elliptic eigenvalue problem; conforming and nonconforming finite element method; two-grid discretization; shifted inverse power method; numerical error estimates; numerical examples