C0 Interior Penalty Methods for Cahn-Hilliard Equations

In this work we study C0 interior penalty methods for Cahn-Hilliard equations. In Chapter 1 we introduce Cahn-Hilliard equations and the time discretization that leads to linear fourth order boundary value problems. In Chapter 2 we review related fundamentals of finite element methods and multigrid methods. In Chapter 3 we formulate the discrete problems for linear fourth order boundary value problems with the boundary conditions of the Cahn-Hilliard type, which are called C0 interior penalty methods, and we carry out the convergence analysis. In Chapter 4 we consider multigrid methods for the C0 interior penalty methods. We present two smoothing schemes and compare their performance. In Chapter 5 we apply the C0 interior penalty methods and the time discretization scheme to nonlinear time-dependent Cahn-Hilliard equations. Numerical examples for phase separation and image processing are presented

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort

Modeling local pattern formation on membrane surfaces using nonlocal interactions

2015 Spring.Includes bibliographical references.The cell membrane is of utmost importance in the transportation of nutrients and signals to the cell which are needed for survival. The magnitude of this is the inspiration for our study of the lipid bilayer which forms the cell membrane. It has been recently accepted that the lipid bilayer consists of lipid microdomains (lipid rafts), as opposed to freely moving lipids. We present two lipid raft models using the Ginzburg-Landau energy with addition of the electrostatic energy and the geodesic curvature energy to describe the local pattern formation of these lipid rafts. The development and implementation of a C⁰ interior penalty surface finite element method along with an implicit time iteration scheme will also be discussed as the optimal solution technique

Code generation for generally mapped finite elements

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations