A two-grid discretization scheme for the Steklov eigenvalue problem

Abstract

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the two-scale scheme is efficient and the approximate solution obtained by the scheme maintains the asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

“…The idea of the two-grid method is related to the ideas in [23,24] for nonsymmetric or indefinite problems and nonlinear elliptic equations. Since then, many numerical 1 methods for solving eigenvalue problems based on the idea of the two-grid method are developed (see, e.g., [5,12,14,28,34,42]). A type of multi-level correction scheme is presented by LinXie [33] and Xie [40].…”

A two-grid method of the non-conforming Crouzeix–Raviart element for the Steklov eigenvalue problem

“…It brings us great difficulties in analyzing the two-grid error estimates. The conclusion in this paper does not follow by using the proof method in [18,22,25] directly. We use the spectral approximation theory and Nitsche-Lascaux-Lesaint technique in space H À 1 2 ð@XÞ and prove that (17) holds (see Lemma 3.1).…”

“…Alonso and Russo [2] and Yang et al [24] discussed the nonconforming finite element approximation of the Steklov eigenvalue problem. Li and Yang [18] studied the two-grid discretization scheme of conforming finite element for the Steklov eigenvalue problem.…”

“…Comparing with [18], this paper analyzes the nonconforming Crouzeix-Raviart element approximation, which has its own feature and some difficulties in theoretical analyzing.…”

Spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem in circular domain

“…There are various approximate methods for solving the Steklov eigenvalue problems. Andreev and Todorov 7 discussed the isoparametric finite element method (FEM); Armentano and Padra 8 introduced and analyzed the conforming linear finite element approximation of the Steklov eigenvalue problem in a bounded polygonal domain; Alonso and Russo, 9 Yang et al, 10 and Li et al 11 studied nonconforming finite elements approximation; Li and Yang 12 and Bi and Yang 13 discussed a 2-grid method of the conforming and nonconforming FEM, respectively; Li et al 14 studied the extrapolation and superconvergence of the Steklov eigenvalue problem; Tang et al 15 studied the boundary element approximation; Bi and Yang 16 discussed the multiscale discretization scheme on the basis of the Rayleigh quotient iterative method; Garau and Morin 17 analyzed the convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems; Cao et al 18 discussed the multiscale asymptotic method for Steklov eigenvalue equations in composite media; Zhang et al 19 discussed the spectral method with the tensor-product nodal basis for the Steklov eigenvalue problem in rectangular domain.…”