半线性抛物方程双网格BDF2虚元格式的误差估计

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2025-07-10 DOI:10.1016/j.apnum.2025.07.002

Peixuan Wu, Xiaohui Wu, Yang Wang, Ruqing Wang

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引用次数: 0

摘要

本文提出了一种新的用虚元法求解半线性抛物方程的两网格离散化方法。在时间维度上考虑了两步后向微分公式（BDF2），而在空间维度上采用了VEM。双网格VEM主要通过在尺寸为H的粗网格上求解非线性系统得到数值解WHn，然后在尺寸为H （H≪H）的细网格上根据先前的结果得到线性系统的数值解WHn。因此，我们的方案不仅减少了总计算费用，而且达到了与单网格VEM相同的精度。具体给出了VEM方法和两网格VEM方法在L2范数和半h1范数上的收敛性分析。

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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Error estimates of a two-grid BDF2 virtual element scheme for semilinear parabolic equation

In this article, we present a new two-grid discretization for the approximation of semilinear parabolic equation found on virtual element method (VEM). The two-step backward differentiation formula (BDF2) is comtemplated in the time dimension, while the VEM is utilized in spatial dimension. The two-grid VEM primarily computes the numerical solution

W_{H}^{n}

from solving a nonlinear system on a coarse mesh with size H and then gets the numerical solution

W_{h}^{n}

to a linear system built by the earlier result

W_{H}^{n}

on a fine mesh with size h (

h ≪ H

). Consequently, our proposed scheme not only reduces total computational expense, but also achieves same accuracy as the single-grid VEM. The convergence analysis in

L^{2}

and semi-

H^{1}

norm for both the VEM and the two-grid VEM methods are provided concretely.

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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.