Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
Adaptive Wavelet/Sparse Grid Solvers for Partial Differential Equations
Participants Frank Koster , Michael Griebel
Keywords Adaptive Grid Refinement, Sparse Grids, Wavelets, Interpolets, Finite Differences, Collocation, Preconditioning 
Description 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 starting-points to cope with this problem:

  • numerical schemes  those work count grows slowly  only with the number of DOF. This may be ac- complished  by e.g.  efficient implementations, multigrid for the linear systems of  equations and so on
  • efficient discretization schemes, where a relatively small number of DOF is sufficient for high accuracy
 Our numerical method combines these two points. A brief introduction is given in the following sections:
  1. Approximation of functions & wavelets
  2. Sparse grids
  3. Discretization
  4. Preconditioners for elliptic linear systems of equations
  5. Numerical example: solution of an elliptic PDE
  6. Numerical example: solution of a convection 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 Navier-Stokes 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)