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.
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