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## Smoothing and h-ellipticity

One of the important concepts to understand with regard to efficient multigrid solvers is that of h-ellipticity Brandt [14]. Elliptic problems have the property that high frequencies are local. That is, if we make a change in the right hand side with a high frequency, the change in the solution is a high frequency change and is localized to the vicinity of the right hand side change. This property follows from the fact that the symbol of the operator defined by
 (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.

Next: Multigrid Approaches for Optimization Up: Review of Multigrid Basics Previous: Full Approximation Scheme (FAS)
Shlomo Ta'asan 2001-08-22