Klaus Deckelnick

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.

Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in

Finite Element Approximation of Dirichlet Boundary Control for Elliptic PDEs on Two- and Three-Dimensional Curved Domains

L^2

A fully discrete numerical scheme for weighted mean curvature flow

to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for the control and the state are obtained in both two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings.

Classical solutions to the Dirichlet problem for Willmore surfaces of revolution

We consider the variational discretization of elliptic Dirichlet optimal control problems with constraints on the control. The underlying state equation, which is considered on smooth two- and three-dimensional domains, is discretized by linear finite elements taking into account domain approximation. The control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions, provided that the underlying mesh satisfies some additional condition. We confirm our analytical findings by numerical experiments.

Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results.

Error analysis of a finite element method for the Willmore flow of graphs

Abstract We consider the Willmore equation with Dirichlet boundary conditions for a surface of revolution obtained by rotating the graph of a positive smooth even function. We show existence of a regular solution by minimisation. Instead of minimising the Willmore functional we reformulate the problem in the hyperbolic half plane and we minimise the corresponding “hyperbolic Willmore functional”.

Finite element approximation of elliptic control problems with constraints on the gradient

The efficient numerical simulation of the curvature-driven motion of interfaces is an important tool in several free- boundary problems. We treat the case of an interface which is given as a graph. The highly non-linear problem is discretized in space by piecewise linear finite elements. Although the problem is not in divergence form it can be written in a variational form which allows the use of the modern adaptive techniques of finite elements. The time discretization is carried out in a semiimplicit way such that in every time step a linear system with symmetric positive matrix has to be solved. Optimal error estimates are proved for the fully discrete problem under the assumption that the time-step size is bounded by the spatial grid size.

Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities

In Theorem 2.2 the choice of the discrete initial value u0h needs to be modified in order to guarantee higher order convergence for ew(0) = ŵh(0) − wh(0). Instead of choosing the initial value u0h as the minimal surface projection û0h of the continuous initial value u0 according to (2.17), we proceed as follows: Let ŵ0h be the solution of (2.22) at time t = 0. We then define the discrete initial value u0h as the solution of the equation ∫

Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.