Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN boundary operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

考虑浸入均匀可压缩空气/流体中的有界、可穿透且各向同性弹性固体对时谐声入射波的散射。通过狄利克雷-诺依曼（DtN）算子，引入精确的透明边界条件，并将模型表述为声-弹相互作用的边值问题。基于对偶论证技术，使用截断 DtN 边界算子导出有限元方法的后验误差估计。后验误差估计由有限元近似误差和 DtN 边界算子的截断误差组成，其中后者相对于截断参数呈指数衰减。提出了一种求解声弹性相互作用问题的自适应有限元算法，其中通过截断误差确定截断参数，并通过有限元离散误差选择用于局部细化的网格单元。数值实验证明了该方法的有效性。