Condition number estimates and weak scaling for 2-level 2-lagrange multiplier methods for general domains and cross points

The 2-Lagrange multiplier method is a domain decomposition method which can be used to parallelize the solution of linear problems arising from partial differential equations. In order to scale to large numbers of subdomains and processors, domain decomposition methods require a coarse grid correction to transport low frequency information more rapidly between subdomains that are far apart. We introduce two new 2-level methods by adding a coarse grid correction to 2-Lagrange multiplier methods. We prove that if we shrink h (the grid parameter) while maintaining bounded the ratio H/h (where H is the size of the subdomains), the condition number of the method remains bounded. We confirm our analysis with experiments on the HECToR (High-End Computing Terascale Resource) supercomputer. This proves that the new methods scale weakly, opening the door to massively parallel implementations.