CADD: A seamless solution to the Domain Decomposition problem of subdomain boundaries and cross-points

The solution of wave problems using Domain Decomposition (DD) requires that the subdomain boundaries should be virtually non-existent, so that waves are not affected by the boundaries. This is a primary problem in DD, and it intensifies in the case of cross-points at which three or more subdomains meet. This topic has received a lot of attention in recent years, with special treatment of cross-points. This paper explains and demonstrates that this problem does not exist in Component-Averaged Domain Decomposition (CADD). CADD is implemented here with the authors’ CARP-CG algorithm, but it is shown that other implementations are also possible. The reason for the non-existence of this problem in CARP-CG is that in some superspace of the problem space, CARP-CG is mathematically equivalent to the CG acceleration of the Kaczmarz algorithm with cyclic relaxation parameters, applied to a single linear system. Due to its advantages, CARP-CG was adopted by some geophysics researchers as the solver of the Helmholtz and the elastic wave equations for full waveform inversion (FWI).