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

Domain Decomposition Methods for Discontinuous Galerkin Approximations of Elliptic Problems

The application of the techniques of domain decomposition to construct effective preconditioners for systems generated by standard methods such as finite difference or finite element methods has been well-researched in the past few decades. However, results concerning the application of these techniques to systems created by the discontinuous Galerkin method (DG) are much more rare. This dissertation represents the effort to extend the study of two-level nonoverlapping and overlapping additive Schwarz methods for DG discretizations of second- and fourth-order elliptic partial differential equations. In particular, the general Schwarz framework is used to find theoretical bounds for the condition numbers of the preconditioned systems corresponding to both the nonoverlapping and overlapping additive Schwarz methods. In addition, the impact on the performance of the preconditioners caused by varying the penalty parameters inherent to DG methods is investigated. Another topic of investigation is the choice of course subspace made for the two-level Schwarz methods. The results of in-depth computational experiments performed to validate and study various aspects of the theory are presented. In addition, the design and implementation of the methods are discussed

Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth

This PhD dissertation concentrates on the development and application of adaptive Discontinuous Galerkin Finite Element (DG-FE) methods for the numerical solution of a Cahn-Hilliard-type diffuse interface model for biological growth. Models of this type have become popular for studying cancerous tumor progression in vivo. The work in this dissertation advances the state-of-the-art in the following ways: To our knowledge the work here contains the first primitive-variable, completely discontinuous numerical implementations of a 2D scheme for the Cahn-Hilliard equation as well as a diffuse interface model of cancer growth. We provide numerical evidence that the schemes above are convergent, with the optimal order. The efficiency of the numerical algorithms depends largely on the implementation of fast solvers for the systems of equations resulting from the DG-FE discretizations. We have developed such capabilities based on multigrid and sparse direct solver techniques. We demonstrate proof-of-concept regarding the implementation of a practical spatially adaptive meshing algorithm for the numerical schemes just mentioned and th1 effective use of a very simple, but powerful, marking strategy based on an inverse estimate. We demonstrate proof-of-concept for a novel simplified diffuse interface model of tumor growth. This model is essentially the Cahn-Hilliard equation with an added source term that is specialized for the context of cancerous tumor progression. We devise and analyze a mixed DG-FE scheme of convex splitting (CS) type for the Cahn-Hilliard equation in any space dimension. Specifically, we prove that our scheme is unconditionally energy stable and unconditionally uniquely solvable. Likewise, we devise and analyze a CS, mixed DG-FE scheme for our diffuse interface cancer model. This scheme is energy stable for any (positive) time step size and for any (positive) space step size that is sufficiently small

Efficient Solvers for the Phase-Field Crystal Equation: Development and Analysis of a Block-Preconditioner

A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation

Adaptive Discontinuous Galerkin Methods for Variational Inequalities with Applications to Phase Field Models

Solutions of variational inequalities often have limited regularity. In particular, the nonsmooth parts are local, while other parts of the solution have higher regularity. To overcome this limitation, we apply hp-adaptivity, which uses a combination of locally finer meshes and varying polynomial degrees to separate the different features of the the solution. For this, we employ Discontinuous Galerkin (DG) methods and show some novel error estimates for the obstacle problem which emphasize the use in hp-adaptive algorithms. Besides this analysis, we present how to efficiently compute numerical solutions using error estimators, fast algebraic solvers which can also be employed in a parallel setup, and discuss implementation details. Finally, some numerical examples and applications to phase field models are presented

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Adaptive Discontinuous Galerkin Finite Element Methods for Second and Fourth Order Elliptic Partial Differential Equations

A unified mathematical and computational framework for implementation of an adaptive discontinuous Galerkin (DG) finite element method (FEM) is developed using the symmetric interior penalty formulation to obtain numerical approximations to solutions of second and fourth order elliptic partial differential equations. The DG-FEM formulation implemented allows for h-adaptivity and has the capability to work with linear, quadratic, cubic, and quartic polynomials on triangular elements in two dimensions. Two different formulations of DG are implemented based on how fluxes are represented on interior edges and comparisons are made. Explicit representations of two a posteriori error estimators, a residual based type and a “local” based type, are extended to include both Dirichlet and Neumann type boundary conditions on bounded domains. New list-based approaches to data management in an adaptive computational environment are introduced in an effort to utilize computational resources in an efficient and flexible manner