The proposed Petrov-Galerkin discretization scheme, while theoretically sound, may face several limitations and challenges in practical large-scale simulations. One significant challenge is the computational complexity associated with the assembly of the discretization matrices. The need to compute integrals over the entire boundary, even when using basis functions with localized support, can lead to increased computational time and memory requirements, particularly for large and complex geometries.
Additionally, the performance of the Petrov-Galerkin method is highly dependent on the stability condition outlined in Assumption 5.2. If this condition is not satisfied for certain geometries or wave numbers, the inf-sup condition may fail, leading to numerical instability and inaccurate results. Furthermore, the choice of boundary element spaces (Raviart-Thomas for the solution space and Buffa-Christiansen for the test space) may not always yield optimal performance across all scenarios, particularly in cases with high-frequency wave propagation where the discretization may struggle to capture the oscillatory nature of the solutions.
Lastly, the scalability of the method can be a concern, as the computational cost may grow significantly with the refinement of the mesh, potentially leading to prohibitive computational times for very fine discretizations required to achieve high accuracy in the solutions.