by John, V..

**Series:** 1994-12, Preprints

- MSC:
- 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 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

estimate,

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.

**Keywords:**