Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann problems than to assemble small Dirichlet problems. Non-overlapping domain decomposition methods such as BDDC (like FETI-DP) can be interpreted as multigrid methods that solve pinned Neumann problems on levels. Unfortunately, the condition number for multilevel BDDC scales as$$C \left(1 + \log \left(\frac{H}{h}\right)\right)^{2L}$$where $L$ is the number of levels and $H/h$ is the coarsening ratio. In contrast, the condition number for multigrid methods with smoothers based on Dirichlet problems have a condition number independent of the number of levels.Is there a way to solve pinned Neumann problems without losing level-independence?
Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels?
pde;multigrid