In the special case of the Laplace operator the vector has the following form

(3.60) |

For a grid with a regular distance between neighboring vertices, the geometrical coefficient matrix yields for five and nine neighboring points:

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

(3.61) |

The vector is reduced to the following form:

(3.62) |

Inserting into the formula (3.58) yields the well known expressions. The function can be written as

For nine points the following formula is obtained:

Michael 2008-01-16