Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity

IF 3 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Scientific Computing Pub Date : 2024-05-02 DOI:10.1137/23m159545x

Jongho Park

引用次数: 0

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1373-A1396, June 2024.
Abstract. We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on [math] and [math] only, where [math] and [math] are the typical diameters of an element and a subdomain, respectively, and [math] measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.

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具有凸能量函数的半线性椭圆问题的加法 Schwarz 方法：收敛率与非线性无关

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1373-A1396 页，2024 年 6 月。摘要。我们研究了具有凸能量函数的半线性椭圆问题的加法施瓦茨方法，这些方法在科学上有着广泛的应用。一个关键的观察结果是，单级和双级加法施瓦茨方法的收敛率都有与问题中的非线性项无关的边界。也就是说，收敛率不会因为非线性的存在而降低，因此解决半线性问题所需的迭代次数并不比线性问题多。此外，两级方法是可扩展的，即该方法的收敛速度只取决于 [math] 和 [math]，其中 [math] 和 [math] 分别是元素和子域的典型直径，[math] 衡量子域之间的重叠。我们提供了数值结果来支持我们的理论发现。

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

SIAM Journal on Scientific Computing 数学-应用数学

CiteScore

5.50

自引率

3.20%

发文量

209

审稿时长

1 months

期刊介绍： The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems. SISC papers are classified into three categories: 1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms. 2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist. 3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.