Adaptive BEM with optimal convergence rates for the Helmholtz equation
Provides theoretical guarantees for adaptive BEM in wave scattering, a known bottleneck for computational acoustics and electromagnetics.
The paper proves that an adaptive boundary element method for the Helmholtz equation achieves optimal convergence rates, independent of the initial mesh, using a residual error estimator.
We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.