HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980205-A

Numerical Experiments

Our numerical results regarding to the Poisson equation with Dirichlet boundary condition given in the weak form: Find $u \in H_0^1(\Omega)$ for a given $f \in L^2(\Omega)$ such that

$$\int_{\Omega} \sum_{i=1}^3 \partial_i u(x) \partial_i v(x... ...nt_{\Omega} f(x) v(x) dx, \quad \forall v \in H_0^1(\Omega),$$ (45)

where $\Omega = \Omega_1 \cup \Omega_2$ and $\Omega_i$ $(i=1,2)$ are unit cubes equivalent to $U$ (see Figure 1.). $\Omega$ is discretized via finite element method using bilinear elements and equidistant grid.

 

Figure 1:

We investigated the effect of the application of the $BPS$ preconditioner [1] and the sparse circulant seminorm representation $C$ - described in the previous section - to the Schur complement part $S$ of the discrete problem corresponding to $\Gamma$. Table 1. shows the change of the condition numbers for different grid sizes $h$. The computation was carried out by MATLAB.

Here we used the following notations:

$BPS^{-1}$ - the inverse of the $BPS$ preconditioner.

$C^{-1}$ - the inverse of $C$.

${\tilde C}^{-1}$ - the approximate inverse of $C$ computed by Chebyshev iteration [11] by $2h^{-1/2}$ iteration steps.

These results show that the use of these matrices in PCG-method to preconditioning $S$ leads to a very fast convergence with contraction factor $q \approx 0.3$. The $BPS$ preconditioner seems to be the best regarding to the contraction factor, however the computation of ${\tilde C}^{-1}$ is cheaper in general, because it costs $O(N^{5/4}log(N))$ arithmetic operation for general $N$ where $N$ denotes the number of unknowns.

$$\vbox{\halign{ \strut \vrule \hfil ...$$

Table 1.

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980205-A