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 $L$ defined by

$$L \exp (i {\bf k} \cdot {\bf x} ) = \hat L ({\bf k}) \exp (i {\bf k} \cdot {\bf x} )$$ (5)

satisfies

$$\vert \hat L( {\bf k)} \vert \geq C \vert {\bf k} \vert ^ {2m},$$ (6)

where ${\bf x} = (x_1, \dots , x_n), {\bf k} = (k_1, \dots , k_n )$. The meaning of this is that the symbol attains high values for high frequencies and small values for low frequencies.

A discrete symbol $\hat L^h (\theta )$ for a discretization $L^h$ of $L$ is defined by

$$L^h \exp (i \theta \cdot {\bf x}/h ) = \hat L^h (\theta ) \exp (i \theta \cdot {\bf x}/h )$$ (7)

where $\theta = (\theta _1, \dots, \theta _n )$, and $\vert\theta \vert = max_j \vert\theta _j\vert \leq \pi$. Discretization of elliptic problems may lead to one of the following analog properties of ellipticity for the symbol of the discretized problem,

$$\mbox{\tt h-ellipticity:} \qquad L^h (\theta) \vert \geq C \sum_{j=1}^d sin ( \theta _j /2)^{2m}$$ (8)

or

$$\mbox{\tt quasi-ellipticity:} \qquad L^h (\theta) \vert \geq C \sum_{j=1}^d sin ( \theta _j )^{2m}$$ (9)

where on the discrete level we consider the discrete transform as

$$\hat U (\theta ) = \sum _\beta \exp ( i \beta \cdot \theta ) U_{\beta}.$$ (10)

In (10) $\beta$ stands for a multi-index and the grid function $U_\beta$ 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

$$L^h = \frac{1}{h^2} \left[ \begin{array}{rrr} & 1 & \\ 1 & -4 & 1 \\ & 1 & \end{array}\right]$$ (11)

has a symbol

$$\hat L^h (\theta ) = -\frac{4}{h^2} ( \sin ^2 (\theta _1 /2) + \sin ^2 (\theta _2/2) )$$ (12)

which is certainly h-elliptic.

quasi-elliptic discretization. The skew Laplacian given by

$$L_x^h = \frac{1}{2 h^2} \left[ \begin{array}{rrr} 1 & & 1 \\ & -4 & \\ 1 & & 1 \end{array}\right]$$ (13)

has a symbol

$$\hat L_x^h (\theta ) = -\frac{2}{h^2} [ \sin ^2 (\theta _1 /2) \cos ^2 (\theta _2 /2) + \sin ^2 ( \theta _2 /2) \cos ^2 (\theta _1 /2)].$$ (14)

Due to the terms $\cos (\theta _1/2), \cos(\theta _2/2)$ this symbol vanishes for the frequency $(\pi , \pi )$ and hence it is not h-elliptic.