Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
5.   Numerical experiment:   solution of an elliptic PDE

We want to solve the Poisson problem

       +    Dirichlet boundary conditions 

The common cycle of sucessive solution of the PDE with a given trial space and the a subsequent refinement depending on the current solution is used. In our case, the current index sets is augmented by neighbours of indices with large wavelet coefficients.

First example:
The right hand side is such that the true solution is u(x,y)=sin(3Pi x)sin(3Pi y) .  In this case, regular sparse grids would be optimal. However, we don't want to use this information, except it tells us how fast the numerical solution can converge, if the refinement strategy and the discretization do not deteriorate the convergence rate. 4th order Interpolets and a 4th order finite difference scheme have been used in this experiment. The figure below shows the numerical error vs. the number of used degrees of freedom. The observed conver- gence rate of 4 is optimal.

Second example:
The right hand side is such that the true solution is u(x,y)=(eps^2 + |x-x0|^2 + |y-y0|^2) ^ (1/4).   This is a regularized variant of a function with a square root singulratity in (x0,y0) \in ]0,1[^2. The regularization parameter eps  was chosen between 10^(-6)  and 0.1 . Again  4th order Interpolets and a 4th order FD scheme were used.  The convergence behaviour is more complicated for this example. When the number of DOF is relatively small the scheme doesn't see the regularization and it converges with a rate of 2 which is optimal. However, for sufficiently large number of DOF and active levels, the regularization comes into play and the convergence rate increases to 4 which is optimal for smooth functions. Unfortunately, the convergence is not very regular. There is much room for improvements.