Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary, to a reaction-diffusion equation on the curve

Abstract

We consider a semi-discrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain Ω ⊂ R 2 , such that the curve meets the boundary ∂Ω orthogonally, and the forcing is a function of the solution of a reaction-diffusion equation that holds on the evolving curve. We prove optimal error bounds for the resulting approximation and present numerical experiments.

Finite element error analysis for a system coupling surface evolution to diffusion on the surface

“…The symmetry of this set-up makes it equatable to the two-dimensional configurations studied in [5] and [13]. In particular we see a travelling wave solution comparable to the ones displayed in Figures 9 and 10 of [5] and Figure 4.4 of [13]. In Figure 4 the initial surface is defined by (5.1) which gives rise to a fully three-dimensional simulation.…”

“…In addition in each plot we display the initial grain boundary, depicted by the blue surface, and the outline of the metallic film A = Ω × [0, 5]. The symmetry of this set-up makes it equatable to the two-dimensional configurations studied in [5] and [13]. In particular we see a travelling wave solution comparable to the ones displayed in Figures 9 and 10 of [5] and Figure 4.4 of [13].…”

“…Using a tangentially modified parametrisation of the evolving curves, [1] obtains error bounds for a corresponding fully discrete scheme. In [13] this idea is applied to the case of open curves Γ(t) meeting a given boundary orthogonally. In each of these papers the error bounds are optimal in H 1 .…”

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

“…For curve shortening flow coupled to a diffusion on a closed curve optimal-order finite element semi-discrete error estimates were shown in [PS17], while [BDS17] have proved convergence of the corresponding backward Euler full discretisation. The case of open curves with a fix boundary was analysed in [SVY20]. For forced-elastic flow of curves semi-discrete error estimates were proved in [PS19].…”