Geometric Error of Finite Volume Schemes for Conservation Laws on Evolving Surfaces

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $\mathbb{R}^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.Comment: 23 pages, 5 figures, to appear in Numerische Mathemati

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

Modelling cell movement and chemotaxis pseudopod based feedback

A computational framework is presented for the simulation of eukaryotic cell migration and chemotaxis. An empirical pattern formation model, based on a system of non-linear reaction-diffusion equations, is approximated on an evolving cell boundary using an Arbitrary Lagrangian Eulerian surface finite element method (ALE-SFEM). The solution state is used to drive a mechanical model of the protrusive and retractive forces exerted on the cell boundary. Movement of the cell is achieved using a level set method. Results are presented for cell migration with and without chemotaxis. The simulated behaviour is compared with experimental results of migrating Dictyostelium discoideum cells

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/maskae/CV_Warwick/Chemotaxis.html

Solving the incompressible surface Navier-Stokes equation by surface finite elements

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori

Error analysis for an ALE evolving surface finite element method

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

Discrete Lie Advection of Differential Forms

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan's homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.Comment: 23 pages, 5 figures, comments welcome! Text slightly modified, references updated in Version 2. Referee comments addresse

Anisotropic diffusion in continuum relaxation of stepped crystal surfaces

We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the surface flux in terms of a new continuum-scale tensor mobility. The starting point is the Burton, Cabrera and Frank (BCF) theory, which offers a discrete scheme for atomic steps whose motion drives surface evolution. Our derivation is based on the separation of local space variables into fast and slow. The model includes: (i) anisotropic diffusion of adsorbed atoms (adatoms) on terraces separating steps; (ii) diffusion of atoms along step edges; and (iii) attachment-detachment of atoms at step edges. We derive a parabolic fourth-order, fully nonlinear partial differential equation (PDE) for the continuum surface height profile. An ingredient of this PDE is the surface mobility for the adatom flux, which is a nontrivial extension of the tensor mobility for isotropic terrace diffusion derived previously by Margetis and Kohn. Approximate, separable solutions of the PDE are discussed.Comment: 14 pages, 1 figur