Adaptivite Schemes for FEM Approximation of the Higher-Order Generalized Cahn-Hilliard Equations

Abstract

The Cahn-Hilliard equation was introduced to model the phase separation in two-component alloys. This is one of the mathematical models most used to describe tumor growth through the evolution of healthy, cancerous and dead cells. In this work we present a numerical study of this equation by introducing higher-order derivatives with a nonlinear source term. Our objective is to solve the higher-order anisotropic problem with locally refined meshes in space using the finite element method for the generalized Cahn-Hilliard model. A phenomenological model that can describe the growth of a cancerous tumor will be treated. Numerical simulations are presented to illustrate the effects of higher-order terms on anisotropy and studies with space adaptivity strategy are also presented indicating its computational efficiency when compared to fixed meshes, especially in the case of anisotropic problems.