  In this paper we propose a finite element method for solving elliptic
equations with the observational Dirichlet boundary data which may subject to
random noises. The method is based on the weak formulation of Lagrangian
multiplier. We show the convergence of the random finite element error in
expectation and, when the noise is sub-Gaussian, in the Orlicz 2- norm which
implies the probability that the finite element error estimates are violated
decays exponentially. Numerical examples are included.
