Convergence of Dziuk's Linearly Implicit Parametric Finite Element Method for Curve Shortening Flow

Convergence of Dziuk's fully discrete linearly implicit parametric finite element method for curve shortening flow on the plane still remains open since it was proposed in 1991, though the corresponding semidiscrete method with piecewise linear finite elements was proved to be convergent in 1994, while the error analysis for the semidiscrete method cannot be directly extended to higher-order finite elements or full discretization. In this paper, we present an error estimate of Dziuk's fully discrete linearly implicit parametric finite element method for curve shortening flow on the plane for finite elements of polynomial degree r ≥ 3. Numerical experiments are provided to support and complement the theoretical convergence result.