Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications

摘要

This paper is dedicated to the control of accuracy and to the adaptivity of the finite element simulation of sound propagation. Assuming time-harmonic behaviour, the mathematical models are given as boundary value problems for the Helmholtz equation. Two singularities inherent to the operator are demonstrated: the k-singularity, related to the phase shift between the exact and the numerical waves, and the A-singularity corresponding to the singularity at the eigenfrequencies. Two a posteriori error estimators are developed and the numerical tests show that, due to these specific singularities, error control cannot, in general, be accomplished by just 'transplanting' methods that work well in static computations. Furthermore, for low wave numbers, it is necessary also to control the influence of the geometric or physical singularities. An h-adaptive version with refinement is applied to 2D and 3D real-life problems.