Guidelines for poisson solvers on irregular domains with dirichlet boundary conditions using the ghost fluid method

Yen Ting Ng, Han Chen, Chohong Min, Frédéric Gibou

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205-227, 2002; 202:577-601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205-227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577-601, 2005) are desirable.

Original languageEnglish
Pages (from-to)300-320
Number of pages21
JournalJournal of Scientific Computing
Issue number2
StatePublished - Nov 2009

Bibliographical note

Funding Information:
Acknowledgements The research of Y.-T. Ng, H. Chen and F. Gibou was supported in part by a Sloan Research Fellowship in Mathematics, by the National Science Foundation under grant agreement DMS 0713858 and by the Department of Energy under grant agreement DE-FG02-08ER15991. The research of C. Min was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-C00045).


  • Ghost fluid method
  • Irregular domains
  • Level set
  • Poisson equation


Dive into the research topics of 'Guidelines for poisson solvers on irregular domains with dirichlet boundary conditions using the ghost fluid method'. Together they form a unique fingerprint.

Cite this