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.

中文翻译：

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非均匀介质中的声散射和不连续梯度问题：有限球法中的分析和 inf-sup 稳定性

在本文中，我们专注于解决时谐声学散射问题的无网格公式，并验证程序的稳定性。声波在非均匀介质中传播，导致不同材料特性区域之间界面上的压力场梯度不连续。无网格方法通常不能准确再现数值解中的不连续性。我们通过引入定义在界面上的拉格朗日乘子场来解决这个问题，以处理压力场梯度的不连续性。该方法不依赖于任何类型的可调参数。我们通过对适用 inf-sup 条件的数值研究表明，由此产生的混合公式会导致适定问题。