 
Summary: A POSTERIORI ERROR ESTIMATES FOR THE STEKLOV EIGENVALUE
PROBLEM
MAR´IA G. ARMENTANO AND CLAUDIO PADRA
Abstract. In this paper we introduce and analyze an a posteriori error estimator for the linear
finite element approximations of the Steklov eigenvalue problem. We define an error estimator of
the residual type which can be computed locally from the approximate eigenpair and we prove
that, up to higher order terms, the estimator is equivalent to the energy norm of the error.
Finally, we prove that the volumetric part of the residual term is dominated by a constant times
the edge residuals, again up to higher order terms.
1. Introduction
The aim of this paper is to propose and analyze an a posteriori error estimator, of the residual
type, for the linear finite element approximations of the Steklov eigenvalue problem.
In recent years, numerical approximation of spectral problems arising in fluid mechanics have
received increasing attention (see [8, 9, 12, 14, 17] and their references). Some of these spectral
problems lead to a Steklov eigenvalue problem similar to the one considered here, for instance,
in the study of surface waves [7], in the analysis of stability of mechanical oscillators immersed
in a viscous fluid ([12] and the references therein) and in the study of the vibration modes of
a structure in contact with an incompressible fluid (see, for example, [9]). In [4] optimal error
estimates for the piecewise linear finite element approximation of the Steklov eigenvalue problem
have been obtained.
