In 2D we use n2 internal nodes and n3 internal nodes in 3D. Based on the PDEs above, this leads to linear systems on the form

It can be shown that the condition numbers of A2 and As are O(h-2); see, e.g., [77]. Since the number of iterations is given by (4.86), we have k = O(%/K) = O(h-1). In 2D, the number of nodes is N « 1/h2 and in 3D, we have N « 1/hs, hence k2 « c2N1/2 in 2D, and ks « csN1/s in 3D. Using < 10-7, we have applied the CG-method to the systems (4.92) and (4.93). The number of iterations are given in Table 4.3 and Table 4.4.

We observe from the tables that the number of iterations is about k2 « 3 N1/2

in 2D and ks « 3.5 N1/s in 3D. Since the amount of work in each iteration is O(N), we have that the solution process requires O(Ns/2) in 2D and O(N4/s) in 3D. The order-optimal result would be O(N) and we will derive methods that are order-optimal in this sense later in the text.

Fig. 4.4. Eigenvalues of the Jacobi iteration Matrix.
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