Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
maximize
next up previous
Next: Computational costs Up: Numerical examples Previous: Numerical examples

Test environment

As discretization we use a five-point finite difference stencil with simple upwinding to obtain stable discretizations already with respect to the finest grid $ \Omega_0$ [21]. This can be interpreted as a discretization which results from a Petrov-Galerkin FE method with operator-dependent trial and test spaces $ {\cal V}_0$ and $ {\cal S}_0$. For reasons of numerical stability we scale the above operators in our computations by a factor $ \varepsilon=1/b$ for $ b\neq0$.

If not stated differently we apply standard multiscale V- or W-cycles or improved multiscale V-cycles together with a complete decomposition of the spaces $ {\cal V}_0$ and $ {\cal S}_0$. Hence, the coarsest grid consists of one interior grid point only. The approximation $ \breve{A}_k^{-1}$ of $ A_k^{-1}$ used in our experiments is computed by a forward- and backward-substitution from an incomplete factorization (ILU(0)) of the blocks $ A_k$ ($ k=1,...,l$).

We start our iterations with a normalized random vector $ u_0^0$ and consider without loss of generality the discrete solution $ u_0=\mathbf{0}$ to a zero right hand side $ f_0=\mathbf{0}$ on the finest grid $ \Omega_0$ [22].

Our measure for the speed of convergence is

$\displaystyle \varrho_{it,1}:=\left(\frac{\vert\vert e_0^{it}\vert\vert _0}{\vert\vert e_0^1\vert\vert _0}\right)^{\frac{1}{it-1}},$ (2.1)

i.e. the average error reduction within $ it$ steps with respect to the discrete $ l^2$-norm [22]. We start to measure after one initial iteration to be free of effects from the starting vector. The stopping criterion is $ 10^{-10}$ for the norm of the absolute error. If this has not been reached with 30 iterations, the iteration is also stopped.


next up previous
Next: Computational costs Up: Numerical examples Previous: Numerical examples
Frank Kiefer
2001-10-25