Theoretical motivation

Inaccurate solution of subproblems reduces the amount of work in each domain decomposition iteration at the cost of some additional work in the outer domain decomposition iteration. Therefore this approach can only lead to a reduction in computing time if the increase in outer domain decomposition iterations is small.

A simple analysis of the condition number of the postconditioned matrix confirms this statement. For symmetric problems, the condition number is a good estimate for the rate of convergence ( for CG) For unsymmetric problems the condition number is less closely linked to convergence.

Each iteration involves solving

with N the matrix from (10). With inaccurate solution of subdomains, we solve a problem

with as in (22) and (21). All subproblems are solved using a relative accuracy.

Theorem 1 relates N and .

Proof:

Proof of (a): Combination of Condition 1 with (inaccurate subdomain solution) gives for all . From the definition of a matrix norm it follows that .
Without loss of generality we take two subdomains, so that N and are described by (15) and (22) respectively. We get

Partition and note that for the Euclidean norm , then we have .
Furthermore, for the Euclidean norm implies and so that finally (a) follows with .
Proof of (b): For any block diagonal matrix , we have: . If we use the additive postconditioner, then is a block diagonal matrix with blocks , so that . Therefore, (b) holds.

Theorem 1 enables us to give a relation between the condition numbers of and .

  

Proof:

Application of Theorem 1, and noting that is a least upper bound norm, gives and .
Since , .
Inequality (27) follows from .

Theorem 2 shows that the subdomain solution accuracy has only a small effect on the condition number of the postconditioned matrix. This means that (at least for symmetric problems) the number of outer iterations will not increase (significantly) when the subdomain accuracy is lowered. The sensitivity of outer loop convergence to is given by the constant C in Theorem 1, which can be chosen 1 for the additive algorithm, independently of the number of subdomains. For multiplicative algorithms this sensitivity constant C will probably also be small and independent of the number of subdomains, however, sharper bounds may require a much more detailed analysis.

The theorems only hold for constant , but the results in Section 6 show that the conclusions also hold in case varies in each iteration.