Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which takes highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with the anisotropic Gibbs-Thomson law with kinetic undercooling, and quasistatic variants thereof. The phase field model is given by. vw(t) + lambda rho(phi)phi(t) = del . (b(phi)del w), c(Psi) a/alpha rho(phi)w = epsilon rho/alpha mu(del phi)phi(t) - epsilon del . A'(del phi) + epsilon(-1)Psi'(phi) subject to initial and boundary conditions for the phase variable. and the temperature approximation w. Here, epsilon > 0 is the interfacial parameter, Psi is a double-well potential, c(Psi) = integral(1)(-1) root 2 Psi(s) ds. rho is a shape function and A(del phi) = 1/2 vertical bar gamma(del phi)vertical bar(2), where gamma is the anisotropic density function. Moreover, v >= 0, lambda > 0, alpha > 0, alpha > 0 and rho >= 0 are physical parameters from the Stefan problem, while b and mu are coefficient functions, which also relate to the sharp interface problem. On introducing the novel fully practical finite element approximations for the anisotropic phase field model, we prove their stability and demonstrate their applicability with some numerical results.