(5) |

satisfies

(6) |

where . The meaning of this is that the symbol attains high values for high frequencies and small values for low frequencies.

A discrete symbol
for a discretization of
is defined by

(7) |

where , and . Discretization of elliptic problems may lead to one of the following analog properties of ellipticity for the symbol of the discretized problem,

(8) |

or

(9) |

where on the discrete level we consider the discrete transform as

(10) |

In (10) stands for a multi-index and the grid function is defined in infinite space.

It is well known that h-ellipticity is required for simple efficient multigrid methods, see Brandt [14]. Actually it can be shown that for such schemes one can construct efficient smoothers. Such smoothers together with a coarse grid correction will result in efficient multigrid solvers. We will come to this when we discuss optimization and the role of h-ellipticity there.

To give the simplest examples for these two types of discretization we take the Laplace equation in two dimensions and the following two discretization

**h-elliptic discretization.**
The standard 5-point discretization of the Laplacian

(11) |

has a symbol

(12) |

which is certainly h-elliptic.

**quasi-elliptic discretization.** The skew Laplacian given by

(13) |

has a symbol

(14) |

Due to the terms this symbol vanishes for the frequency and hence it is not h-elliptic.