For the propagation of elastic waves in unbounded domains, absorbing boundary
conditions at the fictitious numerical boundaries have been proposed. In this paper we
focus on both first- and second-order paraxial boundary conditions(PBCs), which are
based on paraxial approximations of the scalar and elastic wave equations, in the
framework of variational approximations. We propose a penalty function method for
the treatment of PBCs and apply these to finite element analysis. The numerical
verification of the efficiency is carried out through comparing PBCs with viscous
boundary conditions.