Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems

Abstract

The main goal of this article is the study of the convergence and optimality properties of adaptive finiteelement methods (AFEMs) for Steklov eigenvalue problems. These problems arise, for example, in the analysis of the stability of mechanical oscillators immersed in a viscous fluid (see Conca et al., 1995 and the references therein) or in the study of vibration modes of a structure interacting with an incompressible fluid (Bermudez et al., 2000). Finite-element approximations for these problems have been widely used and analysed under a general framework. AFEMs make efficient use of the computational resources, and for certain problems AFEMs are even indispensable for their numerical resolvability. The ultimate goal of adaptive methods is to equidistribute the error and the computational effort to obtain a sequence of meshes with optimal complexity. Historically, the first step to prove optimality has been to understand the convergence of adaptive methods. A basic result for the convergence of linear problems was presented in Morin et al. (2008), where very general conditions on the linear problems and on the ingredients of the AFEM that guarantee convergence were stated. Following these ideas, the (plain) convergence of an AFEM for elliptic eigenvalue problems was proved in Garau et al. (2009). The optimality of an AFEM using Dorfler’s marking strategy (Dorfler, 1996) was proved by Stevenson (2007) and Cascon et al. (2008) for linear elliptic problems. The linear convergence of an AFEM for elliptic eigenvalue problems was proved in Giani & Graham (2009) when considering the approximation of simple eigenvalues, provided that the initial mesh is sufficiently fine, and an improvement of this AFEM including optimality was presented in Dai et al. (2008). In this article we consider the Steklov eigenvalue problem that consists of finding λ ∈ R and u 6≡ 0 such that −∇ ∙ (A∇u)+ cu = 0 in Ω, (A∇u) ∙ −→n = λρu on ∂Ω,