Algebraic Two-Level Convergence Theory for Singular Systems

We consider the algebraic convergence theory that gives its theoretical foundations to classical algebraic multigrid methods. All the main results constitutive of the approach are properly extended to singular compatible systems, including the recent sharp convergence estimates for both symmetric and nonsymmetric systems. In fact, all results carry over smoothly to the singular case, which therefore does not require a specific treatment (except a proper handling of issues associated with singular coarse grid matrices). Regarding problems with a low-dimensional null space, the presented results thus mainly confirm what has been observed often at a more practical level in many previous works. As illustrated by the discussion of the application to the curl-curl equation, the potential impact is greater for problems with large-dimensional null space. Indeed, it turns out that the design of multilevel methods can then actually be easier for the singular case than it is for nearby regularized problems.