Grid Klassenreferenz

a simple adaptive and therefore unstructured grid for 1d problems

This class provides a one dimensional adaptive grid \(\mathcal{T}^{s}=\{e_i\}_{i=0}^{N^s-1}\), where \(s\) indicates how often the grid has been refined and \(\mathcal{T}^{0}\) being the initial grid. The grid entities (subintervals) are numbered consecutively such that the midpoint of the grid can be accessed by the point(int i) method and the width of a grid entity is returned by dx(int i).

For every grid \(\mathcal{T}^{s}\) we consider two mappings

\[ m^{s}:\{0,\ldots,N^{s}-1\} \to \{1,\ldots,N^{s}-1\} \]

and

\[ {m'}^{s}:\{0,\ldots,N^{s}-1\} \to \{1,\ldots,N'^{s}-1\} \]

that map the consecutive numbering of the grid enitities to its memory positions. These mappings are also used for the numbering of the DoFs in our DiscreteFunction implementation. The mappings fulfill the following "sloppy" constraints:

1. The mapping \({m'}^{s}\) is an extension of the mapping \(m^{s-1}\) where grid elements added due to coarsening or refinement are mapped to memory indices above or equal to \(N^{s-1}\). Such this mapping is not continuous, as it contains "holes".
2. During the adaptation phase, we need the old mapping \(m^{s-1}\) as well, in order to find the DoFs of father and children entities where necessary. In this phase, the mappings \(m^{s-1}\) and \({m'}^s\) are both valid mappings and available in the grid's implementation.
3. After the adaptation phase, the grid needs to be compressed via dofCompress, i.e. holes are filled with the newest elements. DiscreteFunction's marked for adaptation need to be changed accordingly, therefore. The only valid and stored mapping, then, is the continuous one \(m^{s}\).

Zu beachten:

Several mappings are valid, if there are no two of those mapping that map any two different entities on the same DoF number.

Only refined and coarsened grid entities cause changes in the DoF storage array.

For the prolonging step of the adaptive algorithm the class provides a mapping \(f^{S}\) called father for all refined entities \(\mathcal{R}^{S}\subset{T}^{S}\), such that

\[ e_i \in \mathcal{R}^{S}, e_j \in \mathcal{T}^{S-1}, e_i \subset e_j \Rightarrow f^{S}(i) = m^{S-1}(j) \]

For the restriction step, the class provides a mapping \(c^{S}\) called children for all entities produced by a coarsening step \(\mathcal{C}^{S} \subset \mathcal{T}^{S}\), such that

\[ e_i \in \mathcal{R}^{S}, e_j, e_k \in \mathcal{T}^{S-1}, e_j \cup e_k = e_i \Rightarrow c^{S}(i) = \{m^{S-1}(j), m^{S-1}(k)\} \]

Beispiele:

test_projection.cc und test_spaceoperator.cc.