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By P. Bastian (auth.), Siegfried Wagner (eds.)

Within the DFG -Schwerpunktprogramm "Stromungssimulation mit Hochleistungsrechnern" and in the actions of the French-German cooperation of CNRS and DFG a DFG symposium on "Computational Fluid Dynamics (CFD) on Parallel structures" used to be geared up on the Institut fur Aerodynamik and Gasdynamik of the Stuttgart college, 9-10 December 1993. This symposium used to be attended by means of 37 scientists. The medical software consisted of 18 papers that thought of finite aspect, finite quantity and a step Taylor­ Galerkin set of rules for the numerical resolution of the Euler and Navier-Stokes equations on vastly parallel pcs with MIMD and SIMD structure and on paintings station clusters. Incompressible and compressible, regular and unsteady flows have been thought of together with turbu­ lent combustion with complicated chemistry. dependent and unstructured grids have been used. excessive numerical potency was once verified by means of multiplicative, additive and multigrid equipment. Shared reminiscence, digital shared reminiscence and dispensed reminiscence platforms have been investigated, at times in line with an automated grid partitioning procedure. a variety of tools for area decomposition have been investigated. the foremost element of those equipment is the solution of the inter­ face challenge as the matrix concerned should be block dense. Multilevel decomposition may be very effective utilizing multifrontal set of rules. The numerical tools contain particular and implicit schemes. within the latter case the approach of equations is usually solved by means of a Gauss -Seidel line re­ laxation technique.

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Found. , 1987. , The Laplacian Spectrum of Graphs, Technical Report, Dep. of Mathematics, University of Ljubljana, Ljubljana, Yugoslavia, 1988. , A Property of Non-Negative Symmetric Matrices and its Application to Graph Theory, Czechoslovak Math. ,25(1975), pp. 619,633. , The Lantzos Algorithm with Partial Reorthogonalization, Math. , Vol. 42, No. 165, pp. 115-142, 1984. , On Estimating the Largest Eigenvalue with the Lantzos Algorithm, Math. , Vol. 38, No. 157,1982. , An Adaptive Finite Element Scheme for Transient Problems in CFD, Compo Meth.

For that, we write (2) Thus, the mesh widths change from hi = 2- i , hi = 2- i . 27r, hk = 2- k from (1) to (3) The constant factors p, kL, kM, kN will be defined for different modifications in the following. In this way, we get a more general notation for the combination method: UL,M,N := :E i+i+k=p+2 Ui,j,k - 2 . :E i+i+k=p+l Ui,i,k + :E i+i+k=p Ui,j,k' (4) Now, i,j, k range from 1 to p, where p has the meaning indicated above. The large indices (like L, M, N) describe mesh widths, and the small ones (like I, m, n) describe exponents of 2.

40-60 24098 Kiel, Germany SUMMARY In this paper we discuss the adaptive solution of parabolic problems. Concentrating on adaptivity in time we describe a time-stepping strategy which is based on extrapolation techniques. The mathematical background in terms of asymptotic expansions and error estimators is presented. The resulting algorithm offers two different sources of parallelism and benefits mainly from basic properties of the time-parallel multigrid method. Numerical results are reported reflecting the sensitivity of the time-stepping strategy.

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