Journal of Scientific Computing

, Volume 54, Issue 2–3, pp 369–413

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries



We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.


Elliptic Parabolic Level-set method Poisson Diffusion Stefan Quadtree Octree Ghost-fluid method 

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Mechanical Engineering Department & Computer Science DepartmentUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Mathematics DepartmentEwha Womans UniversitySeoulSouth Korea
  3. 3.Computer Science DepartmentStanford UniversityStanfordUSA

Personalised recommendations