Parametric finite element method for a nonlocal curvature flow

Abstract

An accurate and efficient parametric finite element method (PFEM) is proposed to simulate numerically the evolution of closed curves under a nonlocal perimeter-conserved generalized curvature flow. We firstly present a variational formulation and show that it preserves two fundamental geometric structures of the flow, i.e., enclosed area increase and perimeter conservation. Then the semi-discrete parametric finite element scheme is proposed and its geometric structure preserving property is rigorously proved. On this basis, an implicit fully discrete scheme is established, which preserves the area-increasing property at the discretized level and enjoys asymptotic equal mesh distribution property. At last, extensive numerical results confirm the good performance of the proposed PFEM, including second-order accuracy in space, area-increasing and the excellent mesh quality.