Immersed Hybridizable Discontinuous Galerkin Method for Multi-Viscosity Incompressible Navier-Stokes Flows on Irregular Domains

Abstract

We present the immersed FFT-based hybridizable discontinuous Galerkin method for solving PDEs with non-smooth solutions on complex geometries including interfaces. We impose jump conditions on the weak formulation of the HDG method to capture discontinuities in the solutions. The solutions near curved interfaces fail to converge optimally due to inappropriate approximation of physical domains bounded by curved boundaries. We employ super-parametric elements in areas connected to high-curvature boundaries to retain optimal convergence rates. We develop a fast solver which is a combination of the FFT and the GMRES for solving linear PDEs. The computation cost is almost linearly proportional to the total number of unknowns in the global system. Finally, we extend the fast solver for solving non-linear incompressible Navier-Stokes equations using the semi-implicit integration in which the influence of the CFL condition on the size of the time step is significantly reduced.