A spectral element solution of the Poisson equation with shifted boundary polynomial corrections: influence of the surrogate to true boundary mapping and an asymptotically preserving Robin formulation (Q6645939)

This paper addresses the computational challenges associated with solving the two-dimensional scalar Poisson equation under general Robin boundary conditions. Such problems frequently arise in practical applications including heat transfer, fluid flow, and wave propagation in acoustics and electromagnetics. The study focuses on high-order methods that accommodate complex geometries without the need for conformal mesh generation, which often poses substantial computational overheads. Specifically, the authors propose an innovative approach integrating a high-order spectral element method (SEM) with a modified shifted boundary method (SBM).\N\NThe proposed method incorporates polynomial corrections for boundary condition enforcement, circumventing the necessity of explicitly evaluating high-order derivatives in the Taylor series expansions typically employed in SBM. Instead, the authors leverage extrapolation and interpolation techniques to project basis functions from the true boundary domain onto a surrogate computational domain. This framework avoids difficulties associated with mesh generation, including challenges introduced by small cut cells, and enables efficient handling of curved domain boundaries.\N\NThe authors employ weak enforcement of Dirichlet, Neumann, and Robin boundary conditions using generalised Nitsche and Aubin methods. Notably, their formulation achieves an asymptotically preserving (AP) approximation for the Robin boundary conditions, ensuring that the numerical solutions respect the asymptotic behaviour of the underlying physical equations. The AP property enables seamless transitions between different types of boundary conditions, enhancing the versatility of the method.\N\NThe methodology is rigorously tested through various numerical experiments, highlighting key performance metrics such as convergence rates, spectral and algebraic conditioning, and accuracy under different polynomial orders and mesh refinements. The experiments confirm the spectral convergence properties of the SEM, as well as the robustness of the polynomial corrections under the SBM framework. The authors also explore the impact of different mappings between the surrogate and true boundaries, underscoring the flexibility of their approach.\N\NThis study's findings have profound implications for computational mathematics, particularly in the efficient and accurate resolution of boundary value problems in complex geometries. By innovating on the traditional SBM and extending its applicability through high-order SEM, the authors offer a computational toolset that balances accuracy, stability, and ease of implementation. This work not only advances the state of numerical methods for PDEs but also provides a foundation for future developments in high-order computational frameworks for scientific and engineering applications.