Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility

1999/01/01 by John W. Barrett, James F. Blowey, Harald Garcke · 7 citations
Materials Science · Computer Science · Mathematics · #Solidification and crystal growth phenomena #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods

paper · doi:10.1137/s0036142997331669

Abstract

We consider a fully practical finite element approximation of the Cahn--Hilliard equation with degenerate mobility \textstyle (∂ u)/(∂ t)= \del .( b(u) \del (-γ\lap u+Ψ'(u))) , where b(⋅)≥ 0 is a diffusional mobility and Ψ(⋅) is a homogeneous free energy. In addition to showing well posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore, an iterative scheme for solving the resulting nonlinear discrete system is analyzed. We also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy. Finally, some numerical experiments are presented.

Citations

• Splitting Algorithms for the Sum of Two Nonlinear Operators
• Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

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• A finite volume/discontinuous Galerkin method for the advective\n Cahn-Hilliard equation with degenerate mobility on porous domains stemming\n from micro-CT imaging

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