A simple finite element method for Stokes flows with surface tension using unfitted meshes. (30th September 2015)

• Record Type: Journal Article
• Authors: Ji, H.; Zhang, Q.
• Is Part Of: International journal for numerical methods in fluids. Volume 81:Number 2(2016)
• Page Start: 87
• Page End: 103
• Publication Date: 2015-09-30
• Subjects: finite element -- discontinuous pressure -- surface tension -- unfitted mesh -- level set function -- Stokes flow; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4176
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: This paper presents a simple finite element method for Stokes flows with surface tension. The method uses an unfitted mesh that is independent of the interface. Due to the surface force, the pressure has a jump across the interface. Based on the properties of the level set function that implicitly represents the interface, the jump of the pressure is removed, and a new problem without discontinuities is formulated. Then, classical stable finite element methods are applied to solve the new problem. Some techniques are used to show that the method is equivalent to an easy‐to‐implement method that can be regarded as a traditional method with a modified pressure space. However, the matrix of the resulting linear system of equations is the same as that of the traditional method. Optimal error estimates are derived for the proposed method. Finally, some numerical tests are presented to confirm the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd. Abstract : For the static three‐dimensional bubble, the proposed method reduces the oscillations near the interface substantially. Note that coefficient matrix of the resulting system of the proposed method is the same as that of the traditional FEM.