Computational Memory

Using domain decomposition we need to solve (factorize) A₁₁, A₂₂ compared to solving A using the standard method. When a direct solver is used, DDM is inherently more memory efficient since a series of smaller matrices are needed to be solved compared to a large matrix. This is evident by assuming that domains are equally partitioned so that N₁ = N₂ = N/2 where N₁, N₂, and N are numbers of unknown for the sub-domains and the original problem respectively. Memory requirement for original problem typically scales as C₁N^a where C₁ is a constant and a > 1. For an M domain decomposition solve, the total memory requirement is

For example, if a = 1.5, using 2-domains we save about 30% of total memory. Also keep in mind that the memory can be distributed across different machines enabling a problem to be solved which is not possible on a single machine.