Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
maximize
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Trial Spaces & Adaptivity

The anisotropic wavelets are used as trial functions for the solution of PDEs or integral equations. Hence, an important task is to select (or to find!) a finite index set ${\cal{T}}$, such that the basis functions

\begin{displaymath}
\{\psi_{\bf l,t}\}_{{\bf l,t} \in {\cal{T}}}
\end{displaymath}

allow for an approximation to the true solution as accurate as posssible. Typically such index sets are computed by a solve-refine cycle, see sections 9.5 and 9.6 for some examples.
However, in some cases one wants to use special, simple trial spaces which require much less implementation effort: uniform and level-adaptive spaces. These correspond to uniform grids and regular sparse grids respectively. We call a trial space uniform, level adaptive or adaptive, if the index set ${\cal{T}}$ is of the following form

uniform ${\cal{T}}$ contains all indices $({\bf l,t})$ for a rectangular block of levels ${\bf l}$ with
$l_i \le L_i$ where $L_i$ is the maximal level along the $i$th coordinate
very simple implementation
level adaptive ${\cal{T}}$ contains all indices $({\bf l,t})$ for an arbitrary set of levels ${\bf l}$ simple implementation
adaptive ${\cal{T}}$ may contain arbitrary indices $({\bf l,t})$ complicated

For algorithmical reasons, there are, however, a few further restrictions on index sets. E.g. so-called cone conditions must hold, see [1,5]. The AWFD user doesn't have to care about this, as all functions for the creation and manipulation of adaptive index sets automatically make sure that the constraints are meet.


next up previous
Next: Discretization of Operators & Up: Mathematical Introduction Previous: Multivariate Wavelets
koster 2003-07-29