Stable Phase Field Approximations of Anisotropic Solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs--Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by {align*} \vartheta\,w_t + \lambda\,\varrho(\varphi)\,\varphi_t & = \nabla \,.\, (b(\varphi)\,\nabla\, w) \,, \cPsi\,\tfrac{a}\alpha\,\varrho(\varphi)\,w & = \epsilon\,\tfrac\rho\alpha\,\mu(\nabla\,\varphi)\,\varphi_t -\epsilon\,\nabla \,.\, A'(\nabla\, \varphi) + \epsilon^{-1}\,\Psi'(\varphi) {align*} subject to initial and boundary conditions for the phase variable $\varphi$ and the temperature approximation $w$. Here $\epsilon > 0$ is the interfacial parameter, $\Psi$ is a double well potential, $\cPsi = \int_{-1}^1 \sqrt{2\,\Psi(s)}\;{\rm d}s$, $\varrho$ is a shape function and $A(\nabla\,\varphi) = \tfrac12\,|\gamma(\nabla\,\varphi)|^2$, where $\gamma$ is the anisotropic density function. Moreover, $\vartheta \geq 0$, $\lambda > 0$, $a > 0$, $\alpha > 0$ and $\rho \geq 0$ 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.