Convergence of a stabilized parametric finite element method of the Barrett–Garcke–Nürnberg type for curve shortening flow

The parametric finite element methods of the Barrett–Garcke–Nürnberg (BGN) type have been successful in preventing mesh distortion/ degeneration in approximating the evolution of surfaces under various geometric flows, including mean curvature flow, Willmore flow, Helfrich flow, surface diffusion, and so on. However, the rigorous justification of convergence of the BGN-type methods and the characeterization of the particle trajectories produced by these methods still remain open since this class of methods was proposed in 2007. The main difficulty lies in the stability of the artificial tangential velocity implicitly determined by the BGN methods. In this paper, we give the first proof of convergence of a stabilized BGN method for curve shortening flow, with optimal-order convergence in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm for finite elements of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the stepsize condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau less-than-or-equal-to c h Superscript k plus 1"> <mml:semantics> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau \leq c h^{k+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (for any fixed constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Moreover, we give the first rigorous characterization of the particle trajectories produced by the BGN-type methods for one-dimensional curves, i.e., we prove that the particle trajectories produced by the stabilized BGN methods converge to the particle trajectories determined by a system of geometric partial differential equations which differs from the standard curve shortening flow by a tangential motion. The characterization of the particle trajectories also rigorously explains, for one-dimensional curves, why the BGN-type methods could maintain the quality of the underlying evolving mesh.