Department of Scientific Computing    Institute for Numerical Simulation    University of Bonn

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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

$$\{\psi_{\bf l,t}\}_{{\bf l,t} \in {\cal{T}}}$$

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.