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Dirichlet boundary conditions

In this section we describe, how functions are computed which take given Dirichlet values. For reasons of simplicity we stick to the two-dimensional case. The extension to higher dimensions is straightforward.

Assume there is a function $v(y)$ of boundary values on the face $\Gamma=\{(1,y)\}$. A particular simple function $\bar{u}(x,y)$ which interpolates $v(y)$ on $\Gamma$ is given by means of the wavelet coefficients $v_{l,t}$ of $v$. Let ${\cal{T}}$ be the set of 2-dimensional indices $({\bf l},{\bf t})$ and ${\cal{T}}_0:=\{ (l_1,t_1): ((L_0,l_1),(2^{L_0} , t_1)) \in {\cal{T}}\}$ the set of (d-1)-dimensional subindices representing the (d-1)-dimensional adaptive grid on $\Gamma$. Then,

$$\bar{u}(x,y)=\sum_{(l,t)\in {\cal{T}}_0} v_{l,t} \psi_{L_0,2^{L_0}}(x) \psi_{l,t}(y) .$$

In the example, $v(y)$ is shown below on the left side and the resulting function $u(x,y)$ is depicted on the right side. The adaptive grid, was a level 5 regular sparse grid.

The function was computed using the member function AdaptiveData<D>::SetBoundaryValueFunction. Up to now, this function is implemented for AdaptiveData<D> only.

In the same fashion as above we can deal with Dirichlet BC on all four faces. In this case we do not only set the coefficients $\bar{u}_{(L_0,.),(2^{L_0},.)}$ but also the coefficients $\bar{u}_{(L_0,.),(0,.)}, \bar{u}_{(.,L_0),(.,0)}, \bar{u}_{(.,L_0),(.,2^{L_0})}$. A resulting function may look like

For more information on how to generate functions with given Dirichlet values, especially the case then the Dirichlet-values on the different faces do not collapse on corners or edges, we refer to the programming reference of AdaptiveData<D>::SetBoundaryValueFunction.