The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov--Poincar\'e operators.

翻译：Neumann-Neumann方法是线性椭圆方程中常用的区域分解方法。然而，该方法在半线性方程中收敛缓慢，且对某些拟线性方程似乎完全不收敛。因此，我们提出两种修正的Neumann-Neumann方法，具有更好的收敛性且计算量更小。我们提供了数值结果，展示了这些方法在半线性和拟线性方程中的应用优势。同时，在方程满足特定假设的条件下，我们证明了网格无关误差缩减的线性收敛性。该分析在一般Lipschitz域上展开，并基于非线性Steklov-Poincaré算子的理论。