Convergence analysis for the Barrett–Garcke–Nürnberg method of transport type for evolving curves

Abstract In this paper, we propose a Barrett–Garcke–Nürnberg (BGN) method for evolving curves under a prescribed background velocity field in $${{\mathbb {R}}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and present the corresponding convergence analysis. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial curves within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.