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$R$-refinement with finite elements or finite
differences for evolutionary PDE models
P A Zegeling
Computational Science Working Group
Dept. of Mathematics, University of Utrecht
P.O. Box 80.010, 3508 TA Utrecht, The Netherlands
e-mail: zegeling@math.ruu.nl
WWW: http://www.math.ruu.nl/people/zegeling/
Abstract
In this talk two different moving-mesh methods ($R$-refinement) are analysed,
compared, and tested on evolutionary PDE models in one and two space
dimensions. The first method (moving finite elements) is based on a
minimization of the PDE residual that is obtained by approximating the
solution by piecewise linear elements. Regularization terms must be
added to prevent the finite-element parametrisation from becoming degenerate.
The second method (moving finite differences) is based on an equidistribution
principle with smoothing both in the spatial and the temporal direction.
Both moving-mesh techniques are based on the method-of lines, which means that
the discretization of the original PDE model and moving-mesh equations is
carried out in two stages. Theory predicts that the finite-element based
moving-mesh method moves its grid points with the flow of the PDE, whereas
the finite-difference based method moves its grid points with the steep
parts of the PDE solution, respectively.
Numerical experiments show results for the finite element and finite
difference case (non-moving and moving) when applied to 1D and 2D
time-dependent models of the convection-diffusion-reaction type.
Different aspects, such as the possibility of grid distortion and influence of
the method parameters, illustrate both the advantages and difficulties of these
methods.
Last modified Fri Jun 21 19:19:17 GB-Eire 1996
(DBD)