Abstract: The Cahn-Hilliard equation, with its wide range of applications in phase-separation, poses numerical challenges due to its non-linear and stiff nature and fourth-order derivatives. This thesis introduces a novel, stabilized, unfitted cut continuous interior penalty finite element method specifically designed for the biharmonic problem with Cahn-Hilliard type boundary conditions, which we have successfully extended to handle the Cahn-Hilliard equation. Our approach combines the theoretical cut discontinuous Galerkin framework for the Poisson problem, as proposed by [1], with the continuous interior penalty biharmonic formulation presented by [2,3]. We prove that this method is well-posed and ensures optimal convergence. The theoretical results are further supported by presented numerical evidence. Finally, we demonstrate the applicability of the method by extending the formulation to handle the Cahn-Hilliard equation using a minimalistic Implicit-Explicit (IMEX) time discretization scheme to manage the non-linearities.
Abstract: This article aims to show the latest methods of mathematical modelling of biological cell membranes. We first presented general research on incorporating multi-physics into mathematical models. We then presented a mathematical and numerical shape optimization framework using a gradient flow method and a finite element method to solve cell membrane dynamics specifically for the elastic bending energy on evolving surfaces.
