Participants  Frank Koster , Michael Griebel  
Keywords  Adaptive Grid Refinement, Sparse Grids, Wavelets, Interpolets, Finite Differences, Collocation, Preconditioning  
Many physical phenomena
are difficult to simulate, because the quantities
of interest contain very complex structures. For example, the
figure below shows the distribution of entropy in a turbulent flow
(simulation by ASCI Turbulence
Project).
In such cases, numerical methods for the solution of the underlying PDE need to spend a very large number of degrees of freedom (DOF) to achieve an accurate approximation to the true solution. This means high demands on computer power and long computing times. There are essentially two startingpoints to cope with this problem:



Links 
Finite Difference Schemes on Sparse Grids for Time dependent Problems Adaptive Parallel Sparse Grids Simulation of Turbulence with Adaptive Methods 

References  [1] R. DeVore Nonlinear Approximation,
Acta Numerica 7 (1998)
[2] M. Griebel and S. Knapek Optimized approximation spaces for operator equations, Constr. Approx. 16 (2000) [3] R. Hochmuth Wavelet bases in Numerical Analysis and Restricted Nonlinear Approximation, Habilitationsschrift FU Berlin (1999) [4] F. Koster Multiskalen basierte Finite Differenzen Verfahren auf adaptiven dünnen Gittern, PhD Thesis Universitsität Bonn (2002) [5] G. Deslauriers and S. Dubuc Symmetric Iterative Interpolation Processes, Constr. Approx. 9 (1989) [6] F. Koster A proof of the Consistency of the Finite Difference Technique on Sparse Grids, Computing 65 (2000) [7] M. Griebel and F. Koster Adaptive Wavelet Solvers for the Unsteady Incompressible NavierStokes Equations in J. Malek, M. Rokyta (Eds.), Advances in Mathematical Fluid Mechanics, Springer Verlag (2000) [8] F. Koster Preconditioners for Sparse Grid Discretizations, Preprint Universität Bonn (2001) 