A Convergent Evolving Finite Element Method with Artificial Tangential Motion for Surface Evolution under a Prescribed Velocity Field

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024.
Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.