Evolving finite element approximations with artificial tangential motion for surface evolution under a prescribed velocity field

Abstract

A novel evolving surface finite element method is proposed to compute the evolution of a closed hypersurface $\Gamma\subset\R^d$, $d=2,3$, moving under a prescribed smooth velocity field $u$. An modified velocity $v$ with an artificial tangential motion is proposed to minimize the instantaneous rate of deformation of the evolving surface, i.e., to minimize the energy functional $\int_{\Gamma}|\nabla_\Gamma v|^2$ under the pointwise constraint $v\cdot n = u\cdot n$, in order to improve the mesh quality of the approximate evolving surface. In order to establish a complete stability and convergence theory for the parametric finite element approximations with artificial tangential motion, we reformulate the problem equivalently in terms of the transport equations of the normal vector $n$ and the second fundamental form $\nabla_\Gamma n$. A novel weak formulation and parametric finite element method are proposed for the reformulated system which couples the modified velocity equation (with artificial tangential motion) with the transport equations of $n$ and $\nabla_\Gamma n$. Optimal-order convergence of the semi-discrete parametric finite element method is proved for finite elements of polynomial degree $k\geq 3$. Numerical examples are presented to illustrate the convergence of the proposed method and the performance of the method in improving the mesh quality of the approximate surface.

Speaker

李步扬博士 2012 年于香港城市大学获得博士学位，先后在南京大学、（德国）图宾根大学、香港理工大学从事科研和教学，现为香港理工大学应用数学系副教授，计算数学杂志 SIAM Journal on Numerical Analysis, Mathematics of Computation, IMA Journal of Numerical Analysis等杂志编委。

主要研究领域为偏微分方程的数值计算和数值分析，包括曲率流的有限元逼近和分析、非线性色散和波动方程不光滑解的计算方法、不可压 Navier–Stokes 方程的计算和分析、高频 Helmholtz 方程的有限元和 PML 方法、非线性抛物方程、相场方程、分数阶偏微分方程、Ginzburg-Landau 超导体方程、热敏电阻方程的数值分析，以及有限元法、谱方法、convolution quadrature等。