 
Summary: Journal of Computational Physics 172, 609639 (2001)
doi:10.1006/jcph.2001.6844, available online at http://www.idealibrary.com on
An Efficient Dynamically Adaptive Mesh
for Potentially Singular Solutions
Hector D. Ceniceros and Thomas Y. Hou
Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106; and
Applied Mathematics, California Institute of Technology, Pasadena, California 91125
Email: hdc@math.ucsb.edu; hou@ama.caltech.edu
Received May 2, 2000; revised May 29, 2001
We develop an efficient dynamically adaptive mesh generator for timedependent
problems in two or more dimensions. The mesh generator is motivated by the vari
ational approach and is based on solving a new set of nonlinear elliptic PDEs for
the mesh map. When coupled to a physical problem, the mesh map evolves with the
underlying solution and maintains high adaptivity as the solution develops compli
cated structures and even singular behavior. The overall mesh strategy is simple to
implement, avoids interpolation, and can be easily incorporated into a broad range
of applications. The efficacy of the mesh is first demonstrated by two examples of
blowingup solutions to the 2D semilinear heat equation. These examples show that
the mesh can follow with high adaptivity a finitetime singularity process. The focus
of applications presented here is however the baroclinic generation of vorticity in
