Acoustic scattering in nonhomogeneous media and the problem of discontinuous gradients: Analysis and inf-sup stability in the method of finite spheres

In this paper we focus on a meshfree formulation for the solution of time-harmonic acoustic scattering problems and verify the stability of the procedure. The sound waves propagate in nonhomogeneous media, giving rise to discontinuities in the gradients of the pressure field across the interfaces between regions of different material properties. Meshfree methods usually do not reproduce accurately the discontinuities in a numerical solution. We overcome this issue by introducing Lagrange multiplier fields defined at the interfaces in order to treat the discontinuities in the gradients of the pressure field. The method does not depend on any kind of adjustable parameter. We show by a numerical study of the applicable inf-sup conditions that the resulting mixed formulation leads to well-posed problems. The use of the proposed method is illustrated in the solution of a number of problems of wave propagation through nonhomogeneous media.