Journal of Scientific Computing

, Volume 54, Issue 2, pp 369-413

First online:

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

  • Frédéric GibouAffiliated withMechanical Engineering Department & Computer Science Department, University of California Email author 
  • , Chohong MinAffiliated withMathematics Department, Ewha Womans University
  • , Ron FedkiwAffiliated withComputer Science Department, Stanford University

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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