Charles M. Elliott

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.

Charles M. Elliott, Chandrasekhar Venkataraman

We consider an arbitrary-Lagrangian-Eulerian evolving surface finite element method for the numerical approximation of advection and diffusion of a conserved scalar quantity on a moving surface. We describe the method, prove optimal order error bounds and present numerical simulations that agree with the theoretical results.

Klaus Deckelnick, Charles M. Elliott, Vanessa Styles

We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a quadratic fidelity term to which we add a perimeter regularisation weighted by a parameter sigma. This yields a functional which is optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. In order to derive a problem which is amenable to computation the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter epsilon. This phase field approach is justified by proving Gamma-convergence to the functional with perimeter regularisation as epsilon tends to zero. The computational approach is based on a finite element approximation. This discretisation is shown to converge in an appropriate way to the solution of the phase field problem. We derive an iterative method which is shown to yield an energy decreasing sequence converging to a discrete critical point. The efficacy of the approach is illustrated with numerical experiments.

Lewis Church, Ana Djurdjevac, Charles M. Elliott

In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fix deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.

Oliver R. A. Dunbar, Charles M. Elliott

We propose a double obstacle phase field methodology for binary recovery of the slowness function of an Eikonal equation found in first traveltime tomography. We treat the inverse problem as an optimization problem with quadratic misfit functional added to a phase field relaxation of the perimeter penalization functional. Our approach yields solutions as we account for well posedness of the forward problem by choosing regular priors. We obtain a convergent finite difference and mixed finite element based discretization and a well defined descent scheme by accounting for the non-differentiability of the forward problem. We validate the phase field technique with a $Γ$ - convergence result and numerically by conducting parameter studies for the scheme, and by applying it to a variety of test problems with different geometries, boundary conditions, and source - receiver locations.

Ana Djurdjevac, Charles M. Elliott, Ralf Kornhuber et al.

In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions.

Till Bretschneider, Cheng-Jin Du, Charles M. Elliott et al.

We present a computational approach for solving reaction-diffusion equations on evolving surfaces which have been obtained from cell image data. It is based on finite element spaces defined on surface triangulations extracted from time series of 3D images. A model for the transport of material between the subsequent surfaces is required where we postulate a velocity in normal direction. We apply the technique to image data obtained from a spreading neutrophil cell. By simulating FRAP experiments we investigate the impact of the evolving geometry on the recovery. We find that for idealised FRAP conditions, changes in membrane geometry, easily account for differences of $\times 10$ in recovery half-times, which shows that experimentalists must take great care when interpreting membrane photobleaching results. We also numerically solve an activator -- depleted substrate system and report on the effect of the membrane movement on the pattern evolution.

Oliver R. A. Dunbar, Charles M. Elliott, Lisa Maria Kreusser

We propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modelled as a wave which emanates from a set of \emph{known} nodes at an initial time, to all other \emph{unknown} nodes at later times with an ordering determined by the arrival time of the information wave front. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial \emph{known} set of nodes to a node is defined as the minimum of a generalised travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each \emph{unknown} node, with boundary conditions at the \emph{known} nodes. The solution value of the local equation at a node is coupled to those of neighbouring nodes with lower values. We provide precise formulations of the model classes and prove equivalences between them. Finally we apply the front propagation models on graphs to semi-supervised learning via label propagation and information propagation on trust networks.

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.

Klaus Deckelnick, Charles M. Elliott, Tatsu-Hiko Miura et al.

We consider the well-posedness and numerical approximation of a Hamilton--Jacobi equation on an evolving hypersurface in $\mathbb R^3$. Definitions of viscosity sub- and supersolutions are extended in a natural way to evolving hypersurfaces and provide uniqueness by comparison. An explicit in time monotone numerical approximation is derived on evolving interpolating triangulated surfaces. The scheme relies on a finite volume discretisation which does not require acute triangles. The scheme is shown to be stable and consistent leading to an existence proof via the proof of convergence. Finally an error bound is proved of the same order as in the flat stationary case.

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.