Step 1 Step

the block size must be O(1= p) can be found in 2]. The observation that the overlap must be O(1= p) was also communicated to the authors by Brandt. To the best of the authors' knowledge, this paper contains the rst published numerical results and analysis. REFERENCES 1] A. Brandt. Multi{level adaptive solutions to boundary{value problems. 9 The results of this analysis agree with Observation 2.1 in the numerical experiments section. Namely, equation(14) shows that relaxing only a segment of line can smooth as eeectively as full line relaxation at points away from the end points of the segment provided that N, the length of the segment, is large enough. However, it may not be an eeective smoother near the endpoints. Therefore, to get eeective smoothing within a block using block-wise line relaxation we must extend the line segments relaxed into the neighboring blocks. Further, equation(18) shows that for small the length of the line segment must be O(1= p) to guarantee eeective smoothing away from the endpoints, i.e. for (j N). This means that block-wise line relaxation can be an eeective smoother provided that both the block size M and the overlap are O(1= p). 4. Conclusions. The result of relating the necessary block size and overlap in a multi-block line relaxation scheme to the strength of the anisotropy that was illustrated by the numerical results in Section 2, appears to hold for many schemes. Some of these may be more eecient on parallel computers than the scheme in Section 2. For example, damped vertical line Jacobi relaxation (with damping parameter ! = 0:7) is known to be an eeective smoother for our model problem (see 4]). Figure (4) illustrates a multi-block version of this relaxation scheme (with block size M = 6 and overlap = 1). Given an initial approximation U, in step 1 a new approximation V is calculated at the indicated points by block relaxation of each of the marked line segments using values of U at all points not on that line segment. In step 2, a new approximation V is again calculated at the indicated points by by block relaxation of each of the marked line segments using values of U at all points not on that line but the value of V calculated in step 1 for points directly above and below the line segment. All line segments in step 1 can be …