Accuracy of post-processing projections for displacement based finite element simulations in room acoustics

In the low-frequency range, time-harmonic room acoustic models are often solved numerically by discretizing the Helmholtz equation with finite element methods, resulting in the scalar acoustic pressure field. An alternative approach is to apply finite element methods to a vector-valued form of the Helmholtz equation, formulated in terms of the Lagrangian displacement field. In this case, computing the acoustic pressure field is required as a post-processing step. The present article focuses on this alternative approach and proposes local post-processing techniques based on Sobolev projections to compute the acoustic pressure from the displacement field solution obtained through a standard finite element method employing Raviart–Thomas discretizations. Projections of varying order and their implementations through weak formulations are demonstrated for continuous and discontinuous Galerkin procedures. The accuracy of these projection techniques is evaluated against the exact analytical solution across different benchmark cases. Additionally, their robustness is measured against noisy displacement data and the computational performance is demonstrated using a realistic auditorium example. The study demonstrates the applicability of the post-processing techniques in room acoustics and suggests that the $H^1$-projection is the most accurate and robust technique among the proposed methods.