## Section: New Results

### A Multilevel Schwarz Preconditioner Based on a Hierarchy of Robust Coarse Spaces

In [32] we present a multilevel preconditioner for SPD matrices. Robust two-level additive Schwarz preconditioners guarantee a fast convergence of the Krylov method. To maintain the robustness each subdomain contributes a small number of vectors to construct a basis for the second level (the coarse space). As long as the dimension of the coarse space is reasonable i.e., direct solvers can be used efficiently, the two-level method scales well. However, the bottleneck arises when factoring the coarse space matrix becomes costly. Using an iterative Krylov method on the second level might be the right choice. Nevertheless, the condition number of the coarse space matrix is typically larger than the one of the first level. One of the difficulties of using two-level methods to solve the coarse problem is that the matrix does not arise from a PDE anymore. We introduce in this paper a practical method of applying a multilevel additive Schwarz preconditioner efficiently. This multilevel preconditioner is implemented in HPDDM and the code for reproducing the results from the paper is available here.