An Analysis of High-Frequency Helmholtz Problems in Domains with Conical Points and Their Finite Element Discretisation

Abstract

Abstract We consider Helmholtz problems in three-dimensional domains featuring conical points. We focus on the high-frequency regime and derive novel sharp upper-bounds for the stress intensity factors of the singularities associated with the conical points. We then employ these new estimates to analyse the stability of finite element discretisations. Our key result is that lowest-order Lagrange finite elements are stable under the assumption that “ ω 2 ⁢ h \omega^{2}h is small”. This assumption is standard and well known in the case of smooth domains, and we show that it naturally extends to domain with conical points, even when using uniform meshes.