Scattering from thin-coated objects: A high-order finite element method

We study the scattering of a time-harmonic acoustic wave by a thinly coated object. The coating is replaced by an e!ective boundary condition of second order, which involves the Laplace–Beltrami operator on the surface of the object. Conditions of this type are also known as Ventcel boundary conditions. For a coating made from a metamaterial with negative e!ective density, the usual “coercive + compact” framework degenerates as the absorption of the coating tends to zero, and it fails for a lossless coating. In this work, we introduce new arguments to show that the problem is well-posed even in the degenerate case. We then analyze its finite element discretization by Lagrange elements of degree $p$ on a curved mesh and show that a quasi-optimal solution under an optimal resolution condition already known in the absence of Ventcell boundary conditions. The second-order boundary condition in the metamaterial regime therefore degrades neither the resolution condition nor the achievable order of convergence.