3. 4. 3 Laplace Operator

In the special case of the Laplace operator the vector $\mathbf{d}$ has the following form

$$\mathbf{d} = [0, 0, 1, 0, 1, 0, \ldots]$$ (3.60)

For a grid with a regular distance $h$ between neighboring vertices, the geometrical coefficient matrix $\mathcal{G}$ yields for five and nine neighboring points:

$$\mathcal{G}_5 = \left[ \begin{array}{c c c c c} 1 & 0 & 0 & 0 & 0... ...2 & 0 \\ 1 & 0 & h & 0 & h^2/2 \\ 1 & 0 & -h & 0 & h^2/2 \end{array}\right]$$

$$\mathcal{G}_9 = \left[ \begin{array}{c c c c c c c c c} 1& ... ... &-2h& 0 & h^2 & 0 & -3h^3/2 & 0 & 8h^4/3 \end{array}\right]$$

After eliminating derivatives which do not appear in the series expansion of the single points, the matrices can be re-written. The derivative vector $[\partial]$ is written as:

$$[\partial] = [u, \partial_x u, \partial_y u, \partial_{xx} u, \pa... ...tial_xxx u, \partial_{yyy} u, \partial_{xxxx} u, \partial_{yyyy} u \ldots] \; .$$ (3.61)

The vector $\mathbf{d}$ is reduced to the following form:

$$\mathbf{d} = [0, 0, 1, 1, 0, \ldots]$$ (3.62)

Inserting into the formula (3.58) yields the well known expressions. The function $K(\mathbf{v}, \mathbf{w})$ can be written as

$$K(\mathbf{w}, \mathbf{v}) := \left\{ \begin{array}{c\vert ... ...} \\ \mathbf{v} \neq \mathbf{w} & h^{-2} \end{array} \right.$$

For nine points the following formula is obtained:

$$K(\mathbf{w}, \mathbf{v}) := \left\{ \begin{array}{c \vert... ...\mathbf{w}, \mathbf{v}) = 2h & -1/12h^{-2} \end{array} \right.$$