The Dirichlet-Neumann (DN) method has been extensively studied for linear partial differential equations, while little attention has been devoted to the nonlinear case. In this paper, we analyze the DN method both as a nonlinear iterative method and as a preconditioner for Newton's method. We discuss the nilpotent property and prove that under special conditions, there exists a relaxation parameter such that the DN method converges quadratically. We further prove that the convergence of Newton's method preconditioned by the DN method is independent of the relaxation parameter. Our numerical experiments further illustrate the mesh independent convergence of the DN method and compare it with other standard nonlinear preconditioners.
翻译:Drichlet-Neumann (DN) 方法已经为线性部分差异方程式进行了广泛研究,但很少注意非线性方程式。 在本文中,我们将DN方法分析为非线性迭代法和牛顿方法的先决条件。我们讨论无主属性,并证明在特殊条件下存在一种放松参数,使DN方法四面形相融合。我们进一步证明,以DN方法为先决条件的牛顿方法的趋同与放松参数无关。我们的数字实验进一步说明了DN方法的网格独立趋同,并将其与其他标准的非线性前提进行比较。