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

We present a new high-order 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, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The 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 Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche's method and (ii) Aubin's method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries.