Finite Element Approximation of a System Coupling Curve Evolution with Prescribed Normal Contact to a Fixed Boundary to Reaction-Diffusion on the Curve

Abstract

We consider a finite element approximation for a system consisting of the evolution of a curve evolving by forced curve shortening flow coupled to a reactiondiffusion equation on the evolving curve. The curve evolves inside a given domain Ω ⊂ R 2 and meets ∂Ω orthogonally. The scheme for the coupled system is based on the schemes derived in [1] and [5]. We present numerical experiments and show the experimental order of convergence of the approximation.

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

“…This approximation forces the boundary conditions to be linear; however, small time steps must be used in order for the end points of the curve to remain close to the boundary , see [6]. In [17] a fully discrete approximation of (10) and (11) in which is presented and a Newton type iteration is used to solve the resulting system of algebraic equations.…”

“…This approximation forces the boundary conditions to be linear; however, small time steps must be used in order for the end points of the curve to remain close to the boundary 𝜕Ω, see [6]. In [17] a fully discrete approximation of ( 10) and (11) in which…”

“…with initial and boundary data We note that for this geometry (17) does not hold. The results are presented in Figure 5, with the left hand plot displaying the evolution of the interface ⃗ X n at t n = 0,1.5,3,4.5,6,7.5, together with the boundary 𝜕Ω (black line) while the right hand plot shows the evolution of the solute concentration W n , plotted against 𝜌, at the same times.…”

“…The analysis in Section 3 also holds if (17) is replaced by |∇F ( p)| = C, p ∈ ∂Ω, for C ∈ R, however we take C = 1 for ease of presentation. We note that from (12) and (13), for any t ∈ [0, T ], we have…”

“…Using the bounds on | x ρ | (18), Sobolev embeddings, (17) and the regularity assumptions ( 12) and ( 16), as well as noting (60), for ρ ∈ {0, 1} and t ∈ [0, T ] we have…”