HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A

  
Basic formulae

We consider the model problem 1 (cf. Figure 1). Following [10] the basis transformation matrix $B_I$ as defined in (3) leads to

$$\begin{eqnarray*}%%\label{My} \mu =\varrho \, (S_C^{-1} T_C^{\phantom{\!\!\!\!-... ...\Vert _{K_{I,i}}^2 } { \Vert{\underline{u}_C} \Vert _{S_C}^2 } \end{eqnarray*}$$

because of the block structure of $\: \overline{M}_I \,=\, \mbox{diag} \{ \overline{M}_{I,i} \}_{i=1,p} \:$, of $\; K_{IC} = K_{CI}^T = [K_{IC,1}\,\ldots,\,K_{IC,p}]$, and of $\: K_I \,=\, \mbox{diag} \{K_{I,i} \}_{i=1,p} \:$ (with $p=2$ subdomains). Introducing

$$\delta_i \::=\: \sup\limits_{\underline{u}_C \in \rm R^{N_C... ...t _{K_{I,i}}^2 } { \Vert {\underline{u}_C} \Vert _{S_C}^2 }$$ (4)

gives $\mu \:\le\: \sum\limits_{i=1}^p \delta_i$. The terms of equation (4) will now be evaluated by means of the Fourier analysis which utilizes the basic ideas of Stüben and Trottenberg [16].

Let the indices $k$ and $i$ correspond to the $x$-directions and $l$ and $j$ to the $y$-direction. Let $n=1/h$ be the (even) number of intervals on $[0,1]$. The eigenvectors and the eigenvalues of the matrix $K_I$ are (cf. [22])

$$\begin{eqnarray*} \mu_{k,l} \: = \: \left\{ \: \mu_k(i) \cdot \mu_l(j) \: \rig... ...\mbox{and}\quad & \lambda_k \:=\: 4 \sin^2 (k \pi h/2) \qquad . \end{eqnarray*}$$

Let the Fourier expansion of $\underline{u}_C$ (on the coupling boundary $\Gamma_C$) be

$$%%\label{uFourier} u_C(j) \;=\; \sum\limits_{l=1}^{n-1} \alp... ...right) \;=\; \sum\limits_{j=1}^{n-1} u_C(j) \mu_l(j) \qquad .$$

Then the denominator of (4) can be approximated as

$$\Vert {\underline{u}_C} \Vert _{S_C}^2 \: = \: \left({ S_... ...1}^{n-1} \sqrt{ \lambda_l^2 + 4\lambda_l\: } \:\alpha_l^2 \\$$ (5)

which is asymptotically exact as $h \to 0$ (cf. Golub [9] and Haase [10]).

The numerator $\Vert {\overline{M}_{I,i}K_{I,i}^{-1}K_{IC,i}\:\underline{u}_C} \Vert _{K_{I,i}}^2$ of (4) has been derived in Haase [10] as follows:

  

$$\underline{y}_I \, := \, K_{IC} \underline{u}_C = \sum\limits_{l=1}^{n-1} \sum\limits_{k=1}^{n-1} \left({\, -\mu_k(n-1) \: \alpha_l \,}\right) \:\mu_{k,l}$$

$$\underline{z}_I \, := \, K_I^{-1}K_{IC}\underline{u}_C \;=\; K_I^{-1}\underline{y}_I = \sum\limits_{l=1}^{n-1} \sum\limits_{k=1}^{n-1} \left({\, \frac{-\mu_k(n-1)}{\lambda_{k,l}} \: \alpha_l \,} \right) \:\mu_{k,l}$$ (6)

$$\Vert {\overline{M}_I K_I^{-1} K_{IC} \: \underline{u}_C} \Vert _{K_I}^2 = \Vert { \overline{M}_I \underline{z}_I } \Vert _{K_I}^2 \;=\; \le... ...ne{M}_I \underline{z}_I \, , \, \overline{M}_I \underline{z}_I \right) \qquad .$$ (7)

With these expansions (5) and (7) the fraction $\delta_i$ and thus the desired spectral radius $\;\mu =\varrho \, (S_C^{-1} T_C^{\phantom{\!\!\!\!-1}})$ will be evaluated. Equation (7) is of special interest since it contains the iteration operator $\overline{M}_I$ which describes the basis transformation $B_I$ (cf. (3)). The next subsection is devoted to the application of the full two-grid operator for the definition of $\overline{M}_I$.

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A