The shifted boundary method for embedded domain computations. Part II: Linear advection-diffusion and incompressible Navier-Stokes equations

We propose a new embedded finite element method for the linear advection-diffusion equation and the laminar and turbulent incompressible Navier-Stokes equations. The proposed method belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. We include the full analysis of stability and convergence of the method in the linear advection-diffusion equation, and a battery of tests for the case of laminar and turbulent incompressible Navier-Stokes equations. We also discuss the conservation properties of the proposed method in all cases.