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A Comparison of Some Error Estimators for Convection-Diffusion Problems on a Parallel Computer.

by John, V..

Series: 1994-12, Preprints

65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65Y05 Parallel computation

Adaptive techniques has been proved to be a necessary tool for the numerical
treatment of partial differential equations. In order to reduce the size
of the arising linear systems more computational work is put only
into subregions where a large local error of the current discrete solution
is assumed. A-posteriori error estimators are used
to choose these subregions.

We consider some convection-diffusion problems in 2d and compare the
results of the adaptive algorithms using different error estimators.
We only choose the mesh size adaptively.
The criteria of comparison
are the reduction of the error in some norm, the time used for doing the
and the parallel behaviour. We only choose such error estimators which work
local and therefore are well fitted for the implementation on a parallel
computer. All error estimators depend on a set of parameters. The comparison
take place with parameters which seems to be reasonable before the solution
process, i.e. we do not compare with 'optimal' sets which are chosen after
having some experience with the test examples.

We present the results of some recent numerical tests on a parallel computer
with a MIMD architecture and 128 processors. No one of the
chosen error estimators was in general superior or interior than the others,
even those which have theoretical shortcomings. All estimators worked
well with general sets of parameters but there is to be expect
a large improvement using other adaptive methods (grid alignment), too.
We found that in general it is not enough only to refine the regions where a
large error is assumed. A certain over-refinement in a neighbourhood of those
regions often yields better results with respect to accuracy of the solution.
Because there is some communication in the estimating process the error
estimators have a modest lost of parallel efficiency.