A CutFEM method for phase change problems with natural convection

In this paper, we present a Nitsche-based CutFEM approach for solving phase change problems while taking natural convection into account in the liquid phase. In the proposed algorithm, the interface is implicitly tracked through a level-set approach, where the level-set function is updated by solving a transport equation. To stabilize the finite element formulation of this equation, we use the continuous interior penalty method (CIP). We use the Backward Euler scheme for the time discretization and the of continuous piecewise linear polynomials (P1) for the spatial discretization of the temperature and the level-set function. For flow velocity and pressure in the liquid phase, the finite element formulation is based on the inf-sup stable pair of Hood-Taylor finite element function spaces P2 - P1. As we are dealing with unfitted finite element methods, stability issues may arise due to geometry discretization. To address this issue, we incorporate ghost penalty terms into the finite element formulations of the heat and Navier-Stokes equations. According to Stefan's condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. For an efficient approximation of this jump, we use a ghost penalty-based domain integral approach which has shown to be highly accurate. To validate the proposed method, we first provide two manufactured solutions that confirm optimal convergence in both temporal and spatial discretizations. We next consider challenging problems such as the melting of n-octadecane in a differentially heated cavity, the multi-cellular melting of gallium, and the freezing of water to further validate our algorithm. We compare our results to existing numerical and experimental data.