Institute for
Numerical Simulation

Research Group
M. Griebel

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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

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