  This paper studies a non-singular coupling scheme for solving the acoustic
and elastic wave scattering problems and its extension to the problems of
Laplace and Lam\'e equations and the problem with a compactly supported
inhomogeneity is also briefly discussed. Relying on the solution representation
of the wave scattering problem, a Robin-type artificial boundary condition in
terms of layer potentials whose kernels are non-singular, is introduced to
obtain a reduced problem on a bounded domain. The wellposedness of the reduced
problems and the a priori error estimates of the corresponding finite element
discretization are proved. Numerical examples are presented to demonstrate the
accuracy and efficiency of the proposed method.
