Hans Fritz

Charles M. Elliott, Hans Fritz

In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.

Hans Fritz

In this paper, we present an algorithm for the computation of harmonic maps, and respectively, of the harmonic map heat flow between two closed Riemannian manifolds. Our approach is based on the totally geodesic embedding of the target manifold into $\mathbb{R}^N$ . Since embeddings of Riemannian manifolds into Euclidean spaces can easily be made totally geodesic by extending the Riemannian metric in a certain way into some tubular neighbourhood, the here presented approach is quite general. Totally geodesic embeddings allow to reformulate the harmonic map heat flow in a neighbourhood of the embedded target manifold. This reformulation has the advantage that the problem becomes unconstrained: Instead of assuming a priori that the solution to the flow maps into the target manifold this fact becomes a property of the solution to the extended flow for special initial data. The solution space to the reformulated problem therefore exists of maps which are also allowed to map into the ambient space of the target manifold. This simplifies the discretization of the problem. Based on this observation, we here propose algorithms for the computation of the harmonic map heat flow and of harmonic maps. In contrast to previous schemes, our algorithm does not make use of projections onto the target manifold, discrete tangential deformations, geodesic finite elements or of Lagrange multipliers. We prove error estimates in the stationary case and present some numerical tests at the end of the paper.

Charles M. Elliott, Hans Fritz, Graham Hobbs

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order surface equations arising in the modelling of biomembranes but the approach may be applied more generally. In particular, we are interested in equations with non-smooth right hand sides and operators which have non-trivial kernels.The theory for well posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

Charles M. Elliott, Hans Fritz

In this paper, we present a general approach to obtain numerical schemes with good mesh properties for problems with moving boundaries, that is for evolving submanifolds with boundaries. This includes moving domains and surfaces with boundaries. Our approach is based on a variant of the so-called the DeTurck trick. By reparametrizing the evolution of the submanifold via solutions to the harmonic map heat flow of manifolds with boundary, we obtain a new velocity field for the motion of the submanifold. Moving the vertices of the computational mesh according to this velocity field automatically leads to computational meshes of high quality both for the submanifold and its boundary. Using the ALE-method in [16], this idea can be easily built into algorithms for the computation of physical problems with moving boundaries.