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: Approximation of functions & wavelets Sparse grids Discretization Preconditioners for elliptic linear systems of equations Numerical example: solution of an elliptic PDE 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)