Neumann-Neumann type domain decomposition of elliptic problems on metric graphs

Journal article, 2025

In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting and rewrite the latter as a preconditioner for the Schur complement system. Then we formulate the discrete iteration as an abstract additive Schwarz iteration and prove that it converges to the finite element solution with a rate that is independent of the finite element mesh size. We also show that the condition number of the Schur complement is uniformly bounded with respect to the finite element mesh size. We provide an implementation and test it on various examples of interest and compare it to other preconditioners.