C1continuous h-adaptive least-squares spectral elementmethod for phase-field models

The phase-field method has been successfully modeled the interface dynamics in multiphase flow phenomena. However, there has been a great disjunction in the interface thickness between in reality and in numerics due to the high gradient of solutions within the interfacial region. By using finer mesh on the interface and coarser mesh in the rest of computational domain, the phasefield method can handle larger scale of problem with realistic length of interface. In this work, a C1 continuous h-adaptive mesh refinement technique with the least-squares spectral element method for the Navier-Stokes-Cahn-Hilliard (NSCH) system and the isothermal Navier-Stokes-Korteweg (NSK) system is presented. Hermite polynomials are used to give global differentiability in the approximated solution, and a space-time coupled formulation and the elementby-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested and they are compared through two numerical examples