Modified Neumann–Neumann methods for semi- and quasilinear elliptic equations

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2024-08-21 DOI:10.1016/j.apnum.2024.08.011

引用次数: 0

Abstract

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 fewer 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–Poincaré operators.

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半线性和准线性椭圆方程的修正诺伊曼-诺伊曼方法

Neumann-Neumann 方法是线性椭圆方程常用的域分解方法。然而，该方法在应用于半线性方程时收敛速度较慢，对于某些准线性方程似乎根本无法收敛。因此，我们提出了两种改进的 Neumann-Neumann 方法，它们具有更好的收敛特性，而且需要的计算量更少。我们提供的数值结果显示了这些方法在应用于半线性方程和准线性方程时的优势。我们还证明了在方程的某些假设条件下，与网格无关的误差减少的线性收敛性。分析是在一般 Lipschitz 域上进行的，并依赖于非线性 Steklov-Poincaré 算子理论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.