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 convergences to the finite element solution with a rate that is independent of the finite element mesh size. We show that the condition number of the Schur complement is also independent of the finite element mesh size. We provide an implementation and test it on various examples of interest and compare it to other preconditioners.

翻译：本文针对度量图上的椭圆问题，提出了一种Neumann-Neumann型区域分解方法。我们描述了连续与离散情形下的迭代格式，并将离散迭代重新表述为Schur补系统的预条件子。随后，将离散迭代形式化为抽象加性Schwarz迭代，并证明其以与有限元网格尺寸无关的收敛速率收敛至有限元解。我们证明了Schur补的条件数同样与有限元网格尺寸无关。我们提供了算法实现，并在多种典型算例上进行了测试，同时与其他预条件子进行了比较。