Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces - Elliott, Charles M.; Garcke, Harald; Kovács, Balázs (Numerische Mathematik)
Abstract An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $$H^1$$ H1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
Keywords: Applied Mathematics; Computational Mathematics

Time-dependent electromagnetic scattering from thin layers - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Abstract The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces - Beschle, Cedric Aaron; Kovács, Balázs (Numerische Mathematik)
Abstract In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

Correction to: Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

A convergent evolving finite element algorithm for Willmore flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Linearly implicit full discretization of surface evolution - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Radial basis function approximation of noisy scattered data on the sphere - Hesse, Kerstin; Sloan, Ian H.; Womersley, Robert S. (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Convergence of finite elements on an evolving surface driven by diffusion on the surface - Kovács, Balázs; Li, Buyang; Lubich, Christian; Power Guerra, Christian A. (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Discrete variational Lie group formulation of geometrically exact beam dynamics - Demoures, F.; Gay-Balmaz, F.; Leyendecker, S.; Ober-Blöbaum, S.; Ratiu, T. S.; Weinand, Y. (Numerische Mathematik)

A subdivision algorithm for the computation of unstable manifolds and global attractors - Dellnitz, Michael; Hohmann, Andreas (Numerische Mathematik)