Journal of Scientific Computing

, Volume 54, Issue 2, pp 369–413

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

Authors

    • Mechanical Engineering Department & Computer Science DepartmentUniversity of California
  • Chohong Min
    • Mathematics DepartmentEwha Womans University
  • Ron Fedkiw
    • Computer Science DepartmentStanford University
Article

DOI: 10.1007/s10915-012-9660-1

Cite this article as:
Gibou, F., Min, C. & Fedkiw, R. J Sci Comput (2013) 54: 369. doi:10.1007/s10915-012-9660-1

Abstract

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.

Keywords

EllipticParabolicLevel-set methodPoissonDiffusionStefanQuadtreeOctreeGhost-fluid method

Copyright information

© Springer Science+Business Media New York 2012