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

  • Authors:

  • Frédéric Gibou;Chohong Min;Ron Fedkiw

  • Affiliations:

  • Mechanical Engineering Department & Computer Science Department, University of California, Santa Barbara, USA 93106;Mathematics Department, Ewha Womans University, Seoul, South Korea 120-750;Computer Science Department, Stanford University, Stanford, USA 94305-9020

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2013

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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.