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   -> Volume 11, Issue 1

Preprint: Wavelets for PDEs and 3D Turbulence Simulations
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Frank Koster (

PostPosted: Wed Jan 02, 2002 6:06 pm    
Subject: Preprint: Wavelets for PDEs and 3D Turbulence Simulations
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Preprint: Multiscale Methods for the Solution of the Navier-Stokes Equations
M. Griebel, F. Koster

In this paper we apply the finite difference method on adaptive sparse grids to the simulation of turbulent flows. This method combines the flexibility and efficiency of finite difference schemes with the advantages of an
adaptive approximation by tensor product multiscale bases. We shortly discuss the method. Then, we present numerical results for a simple linear convection problem for a validation of our scheme. Finally, results for three-dimensional turbulent shear layers are shown.

Preprint: Preconditioners for Sparse Grid Discretizations
F. Koster

In this paper we deal with preconditioners for sparse grid finite-difference-- and Petrov-Galerkin--discretizations of the Poisson equation. We analyse the Jacobi-preconditioner for the simple setting of non-adaptive grids and periodic boundary conditions. The analysis shows that the resulting condition numbers mainly depend on the underlying tensor product Wavelets. For example, high order Lifting-Interpolets lead to $l_2$-condition numbers which are essentially independent of the finest mesh size. Based on this observation we introduce a so-called Lifting-preconditioner for discretizations which use Interpolets as trial-functions. Numerical examples show the efficiency of the preconditioners for cases which are not covered by our analysis, e.g., adaptive grids.

More stuff on adaptive sparse grid discretizations can be found in my thesis ( in german only )

Best regards
Frank Koster
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