A Comparison of Some Error Estimators for Convection-Diffusion Problems on a Parallel Computer.
by John, V.
Preprint series: 94-12, Preprints
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 65Y05 Parallel computation
Abstract: Adaptive techniques has been proved to be a necessary tool for the numericaltreatment of partial differential equations. In order to reduce the size of the arising linear systems more computational work is put onlyinto subregions where a large local error of the current discrete solutionis assumed. A-posteriori error estimators are usedto choose these subregions. We consider some convection-diffusion problems in 2d and compare theresults 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 estimate,and the parallel behaviour. We only choose such error estimators which worklocal and therefore are well fitted for the implementation on a parallelcomputer. All error estimators depend on a set of parameters. The comparisontake place with parameters which seems to be reasonable before the solutionprocess, i.e. we do not compare with \'optimal\' sets which are chosen afterhaving some experience with the test examples.We present the results of some recent numerical tests on a parallel computerwith a MIMD architecture and 128 processors. No one of thechosen error estimators was in general superior or interior than the others,even those which have theoretical shortcomings. All estimators workedwell with general sets of parameters but there is to be expecta large improvement using other adaptive methods (grid alignment), too.We found that in general it is not enough only to refine the regions where alarge 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 errorestimators have a modest lost of parallel efficiency.
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