We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

翻译：我们提出了一种适用于一般Robin边界条件的二维标量泊松方程的高阶精确谱元解法。该解法基于移边界方法的简化版本，采用连续任意阶$hp$-Galerkin谱元法作为数值离散过程。其简化核心在于通过多项式校正避免显式计算泰勒级数展开中的高阶偏导数——而后者在传统移边界方法中必不可少。在此框架下，我们采用外推法与新型插值法，将基函数从真实域投影到近似代理域。由此得到的解法能自然处理域的曲线几何特征、规避复杂繁琐的网格生成过程，并避免小切割单元问题。Dirichlet、Neumann及一般Robin边界条件通过以下两种弱形式施加：i）广义Nitsche方法和ii）广义Aubin方法。为此，我们提出了嵌入式Robin公式的一致渐进保持形式。通过数值实验与算法特性分析，我们研究了不同弱形式的性能，包括多项式$p$（基函数阶次增加）、网格$h$（细化）及矩阵条件数下的收敛性分析，分别突出谱收敛与代数收敛特征。这些工作旨在评估变分形式、多项式阶次、网格尺寸以及真实边界与代理边界间映射对误差的影响。