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Copyright:
1996, 1999, 2006
J.A. Sethian

Adaptive Mesh Refinement for Level Set Methods
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In some cases, one wishes to employ adaptive mesh refinement in the calculation of the evolving interface. This is most often done in the context of adaptive mesh refinement for the underlying physics, such as in fluid flow solvers. The technique for coupling this to level set methods in the case of speed functions which do not intimately depend on curvature is fairly straightforward, using much of the adaptive mesh methodology built over the past decade and a half.


Annotated References:

The first work on adaptive mesh refinement in the context of level set methods was done by Milne in Ref. 1, who did so in his PhD. thesis at Berkeley. Milne showed that the handling of the terms associated with the hyperbolic transport part of the equation of motion was straightforward, and demonstrated results in two and three space dimensions. In the case of curvature-driven flow, the techniques are considerably more subtle, and require a delicate attention to the fine grid/coarse grid boundary. Unless care is taken, this fine/coarse boundary can act as a source or sink, and greatly reduce the accuracy of the calculation.

Since this work, a variety of adaptive level set methods have become available.
  • Ref. 2 shows the role of adaptivity in both level set and Fast Marching Methods.
  • Ref. 3 discuss parallel implementation of interface techniques, including both message-passing architectures and global programming languages.


New Book and Resource on Level Set and Fast Marching Methods



References:

  1. Adaptive Level Set Methods Interfaces : Milne, B., PhD. Thesis, Dept. of Mathematics, University of California, Berkeley, CA., 1995.
  2. Adaptive Fast Marching and Level Set Methods for Propagating Interfaces : Sethian, J.A., Acta Numerica, Proceedings of ALGORITMY '97, Zuberec, 1997.
    Abstract

    Adaptivity provides a way to construct optimal algorithms for tracking moving interfaces which arise in a wide collection of physical applications. Here, we summarize the development and interconnection between Narrow Band Level Set Methods and Fast Marching Methods, which provide efficient techniques for tracking fronts. We end with a small collection of examples to demonstrate the applicability of the techniques.

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  3. Parallel Level Set Methods for Propagating Interfaces on the Connection Machine : Sethian, J.A., Unpublished manuscript, Thinking Machines Corporation, 1989.
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Additional References:
  1. An adaptive level set approach to incompressible two phase flow : Sussman, M., Almgren, A., Bell, J., Colella, P., Howell, L., and Welcome, M., 148, 1, pp. 81-124, 1999,