Gerhard Dziuk

An algorithm for evolutionary surfaces

SummaryAn Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateaus problem forH-surfaces.

Computation of geometric partial differential equations and mean curvature flow

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Finite element methods for surface PDEs

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

A finite element method for surface restoration with smooth boundary conditions

In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C1-continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L2-gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as a system of two coupled second order equations. As it is well known, fourth order problems require an implicit time discretization. Here a semi-implicit approach is presented which allows large time steps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.

Evolution of Elastic Curves in

We consider curves in

\Rn

{\mathbb R}^n

: Existence and Computation

moving by the gradient flow for elastic energy, i.e., the L2 integral of curvature. Long-time existence is proved in the two cases when a multiple of length is added to the energy or the length is fixed as a constraint. Along these lines, a lower bound for the lifespan of solutions to the curve diffusion flow is observed. We derive algorithms for both the elastic flows and the curve diffusion equation. After a numerical test we compute several examples, including cases of curve diffusion in which a singularity develops.

An Adaptive Finite Element Method for the Laplace-Beltrami Operator on Implicitly Defined Surfaces

We present an adaptive finite element method for approximating solutions to the Laplace-Beltrami equation on surfaces in

Computational parametric Willmore flow

\mathbb{R}^3

CONVERGENCE OF A SEMI-DISCRETE SCHEME FOR THE CURVE SHORTENING FLOW

which may be implicitly represented as level sets of smooth functions. Residual-type a posteriori error bounds which show that the error may be split into a “residual part” and a “geometric part” are established. In addition, implementation issues are discussed and several computational examples are given.