An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

Xiaoming Zheng, John Lowengrub

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.

Original languageEnglish
Pages (from-to)1225-1257
Number of pages33
JournalAdvances in Computational Mathematics
Volume42
Issue number5
DOIs
StatePublished - Oct 1 2016

Keywords

  • Adaptive mesh
  • Curvature
  • Elliptic equations with jump conditions
  • Free interface problems
  • Interface-fitted
  • Level-set method
  • Stokes equations
  • Surface tension
  • Tumor growth

Fingerprint

Dive into the research topics of 'An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension'. Together they form a unique fingerprint.

Cite this