High resolution sharp computational methods for elliptic and parabolic problems in complex geometries

Frédéric Gibou, Chohong Min, Ron Fedkiw

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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.

Original languageEnglish
Pages (from-to)369-413
Number of pages45
JournalJournal of Scientific Computing
Issue number2-3
StatePublished - Feb 2013

Bibliographical note

Funding Information:
Acknowledgements The research of F. Gibou was supported in part by ONR N00014-11-1-0027, NSF CHE 1027817, DOE DE-FG02-08ER15991, ICB W911NF-09-D-0001 and by the W.M. Keck Foundation. The research of C. Min was supported in part by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0028298) and by the Korea Research Foundation Grant funded by the Korean Government (KRF-2011-0013649). The research of R. Fedkiw was supported in part by ONR N00014-09-1-0101, ONR N-00014-11-1-0027, ARL AHPCRC W911NF-07-0027, NSF IIS-1048573, and the Intel Science and Technology Center for Visual Computing.


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


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