The shifted boundary method for embedded domain computations using a high-order Spectral Element method for the 2D Poisson problem

In recent years, the Shifted Boundary Method (SBM) have gained interest due to the its ability to handlecomplex domains through embedded domain computations. The SBM address the problem of avoidingsmall cut cells and makes the meshing task close to trivial. The key feature - and the expense - ofthe SBM is how the boundary conditions (BCs) are applied on a surrogate/approximate boundary viathe use of Taylor expansions to ensure that the convergence rates of the overall discrete formulation ispreserved, see [1,2]. This original work by Main & Scovazzi was exploiting the classical - second-orderaccurate - Finite Element Method (FEM), however, higher-order contributions have recently been made,see [3,4]. One high-order numerical method is the Galerkin-formulated Spectral Element Method (SEM)[5] that can be viewed as a multi-domain version of the single-domain polynomial spectral method.We present a SEM-based model combined with the SBM for solving the Poisson equation in 2D ondifferent domains/geometries with various BCs. Convergence studies are performed to establish thelegitimacy of the work, including considerations of matrix conditioning when imposing Dirichlet BCsweakly via Nitsche’s variational form. Also, the presented work is free of higher-order derivatives fromthe aforementioned Taylor expansions, as the formulation utilizes the polynomial behaving nature of thebasis functions