Pure and Applied Mathematics Quarterly

Volume 14 (2018)

Number 1

Special Issue: In Honor of Chi-Wang Shu

Guest Editors: Jian-Guo Liu and Yong-Tao Zhang

A weak formulation for the multiphase Stokes flow problem without body fitting grids

Pages: 131 – 159

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n1.a5


Ningchen Ying (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

Songming Hou (Department of Mathematics & Statistics and Center of Applied Physics, Louisiana Tech University, Ruston, La., U.S.A.)

Shingyu Leung (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong)

Hongkai Zhao (Department of Mathematics, University of California at Irvine)


We develop an effective interface tracking method to simulate the incompressible Stokes flow with moving interfaces. The Stokes equations are first rewritten into a system of elliptic equations with singular sources which can be efficiently solved by a simple weak formulation proposed in [11]. The key idea is to first split the solution into a singular part and a regular part additively. The singular part captures the interface conditions, while the regular part approximates the equations in the whole domain, which can be solved by the standard finite element formulation. We carefully design numerical methods to interpolate the velocity to the moving interface. Numerical tests are carried out to demonstrate the accuracy and other properties of our method.

S. Hou’s research is partially supported by the Walter Koss Endowed Professorship. This professorship is made available through the State of Louisiana Board of Regents Support Funds. The work of Leung was supported in part by the Hong Kong RGC grants 16303114 and 16309316. The research of Zhao is partially supported by NSF grant DMS-1418422.

Received 4 December 2017

Published 2 April 2019