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Choice of Coarse Grid Size

Given , it has been observed empirically (see Gropp and Keyes [110]) that there often exists an optimal value of which minimizes the total computational time for solving the problem. A small provides a better, but more expensive, coarse grid approximation, and requires solving more, but smaller, subdomain solves. A large has the opposite effect. For model problems, the optimal can be determined for both sequential and parallel implementations (see Chan and Shao [51]). In practice, it may pay to determine a near optimal value of empirically if the preconditioner is to be re-used many times. However, there may also be geometric constraints on the range of values that can take.