Finite Element Structure Preserving Numerical Approximations Of Thermodynamically Consistent Crystal Growth Models

We reformulate the thermodynamically consistent phase field model for dentritic crystal growth(i.e.the model satisfying the first law and second law of thermodynamics)with the Energy Quadratization(EQ)method and Scalar Auxiliary Variable(SAV)method,then design two fully discrete numerical schemes.The two schemes are both second order in time,linear and energy and entropy production rate preserving for any time steps.Note that SAV method is a special version of EQ method.The SAV scheme is effectively decomposed into a series of Poisson equations in numerical implementation.The two schemes are first discretized in time aided by the Crank-Nicolson finite difference method and then in space using a second order finite element methods.The linear systems resulting from the schemes are shown to be uniquely solvable at both the semi-discrete and the fully discrete level.Mesh refinement tests are performed to show the second-order time convergence rate in the schemes.Several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the schemes.The effects of various model parameters on growth patterns of the crystal growth are further investigated in details with the numerical solver.The approach to developing the energy and entropy production rate-preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models.