Fast Adaptive Methods for the Free-Space Heat Equation (Strain)

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SIAM Journal on Scientific Computing
Volume 15-1, January 1994, pp.  185-206
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved


Title:            Fast Adaptive Methods for the Free-Space Heat 
                  Equation

Author:           John Strain

AMS Subject
Classifications:  65M50, 33A65, 35K05, 80A20, 65V05, 65R10

Key words:        heat equation, heat potentials, fast algorithms, 
                  adaptive methods, crystal growth

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ABSTRACT

Standard numerical methods for the heat equation in two or more space 
dimensions are excellent if it is necessary to follow the evolution 
in great detail through many small timesteps. This paper presents 
efficient and accurate new adaptive methods that solve the free-space 
heat equation with {\em large\/} timesteps.  These methods combine the 
fast Gauss transform with an adaptive refinement scheme that 
represents the solution as a continuous piecewise polynomial, to a 
user-specified degree of accuracy. The same approach is extended to 
solve inhomogeneous problems and to solve the heat equation in moving 
domains with boundaries. In problems with boundaries, it allows the 
use of accurate boundary representations without requiring difficult 
product integration formulas or precluding fast evaluation schemes. 
Numerical experiments in two space dimensions show these methods to be 
accurate and efficient, especially for highly nonuniform or 
discontinuous initial data or when substantial accuracy is required.


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