A multigrid finite element solver for the Cahn-Hilliard equation
Journal of Computational Physics
Effective condition number of the Hermite finite element methods for biharmonic equations
Applied Numerical Mathematics
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Phase-field models with conserved phase-field variables result in a 4th order evolution partial differential equation (PDE). When coupled with the usual 2nd order thermo-mechanics equations, such problems require special treatment. In the past, the finite element method (FEM) has been successfully applied to non-conserved phase fields, governed by a 2nd order PDE. For higher order equations, the convergence of the standard Galerkin FEM requires that the interpolation functions belong to a higher continuity class. We consider the Cahn-Hilliard phase-field model for diffusion-controlled solid state phase transformation in binary alloys, coupled with elasticity of the solid phases. A Galerkin finite element formulation is developed, with mixed-order interpolation: C^0 interpolation functions for displacements, and C^1 interpolation functions for the phase-field variable. To demonstrate convergence of the mixed interpolation scheme, we first study a one-dimensional problem - nucleation and growth of the intermediate phase in a thin-film diffusion couple with elasticity effects. Then, we study the effects of completeness of C^1 interpolation on parabolic problems in two space dimensions by considering the growth of the intermediate phase in a binary system. Quadratic convergence, expected for conforming elements, is achieved for both one- and two-dimensional systems.