A First Course in the Numerical Analysis of Differential Equations

13

Multigrid techniques

13.1  In lieu of a justification . . .

How good is the Gauss–Seidel iteration (12.21) at solving the five-point equations on an m × m grid? On the face of it, posing this question just after we have completed a whole chapter devoted to iterative methods is neither necessary nor appropriate. According to (12.51), the spectral radius of the iteration matrix is cos²[π/(m + 1)] ≈ 1 − π²m⁻² and inspection of the third row of Fig. 12.5 will convince us that this presents a fair estimate of the behaviour of the scheme. Yet, by ...