The high-order Shifted Boundary Method and its analysis

The Shifted Boundary Method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods, that has proven efficient in handling partial differential equation problems with complex geometries. The key feature of the SBM is a shift in the location where boundary conditions are applied - from the true to a surrogate boundary - and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. This paper presents the high-order version of the method, its mathematical analysis, and numerical experiments. The proposed method retains optimal accuracy for any order of the finite element interpolation spaces despite the surrogate boundary is piecewise linear. As such, the proposed approach bypasses the problematic issue of meshing complex geometries with high-order body-fitted boundary representations, without the need of complex data structures for the integration on cut cells.