A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering

Abstract

We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And the coefficient matrices associated with those linear systems are constructed to be symmetric and positive definite. We prove error estimates of eigenvalues and eigenfunctions. Numerical results coincide in theoretical analysis and indicate our scheme is highly efficient in solving the eigenvalue problem.

A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

“…We can use the algorithm in Remark 4.1 in [47] to compute u ⊥ h and then obtain u * h . Lemma 2.4 ( [46,51]). Assume (λ, u) ∈ C × V and (λ, u * ) ∈ C × V satisfy (2.2) and (2.4), respectively, and suppose w, w * ∈ V such that b(w, w * ) = 0.…”

“…Recently, [44] propose a parallel multilevel correction method for linear selfadjoint eigenvalue problems. Especially, multilevel correction method has been applied to non-selfadjoint Steklov eigenvalue problems in [51]. As we know, the multigrid method [3,7,8,20] as an efficient preconditioners provide an optimal order algorithm for solving boundary value problems.…”

“…Hence, the aim of this paper is to present a full multigrid method [15,26,44] (sometimes also referred to as nested finite element method) for solving non-selfadjoint Steklov eigenvalue problems based on the combination of the multilevel correction method and the multigrid iteration for boundary value problems. Comparing with the method in [29,36,37,51], we do not need to solve the linear boundary value problem exactly in each correction step in this paper. Some multigrid iteration steps are used to get an approximate solution.…”

A locking‐free shifted inverse iteration based on multigrid discretization for the elastic eigenvalue problem

“…The multigrid or multilevel discretization is an efficient tool for eigenvalue problems (e.g., see previous studies 43–49 ). Gong et al 43 developed a shifted inverse adaptive multigrid method for the elastic eigenvalue problem.…”

The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering

“…Many studies have been carried out on numerical methods for the Steklov eigenvalue problems (see, e.g., [6,[8][9][10][11][12][13][14][15][16][17][18][19]). In recent years, researchers have paid attention to the Steklov eigenvalue problem in inverse scattering (see, e.g., [7,[20][21][22]).…”