Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

IF 1.9 3区 数学 Q2 Mathematics
R. Eymard, David Maltese
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引用次数: 1

Abstract

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a  linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods  show convergence properties to a continuous solution of the problem in a weak sense, through the change  of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued  in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a  weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D  and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate  results. We then construct a numerical scheme which presents a convergence property to the entropy  weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing  to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation,  and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term.
具有不规则数据的椭圆型线性问题非线性数值逼近的收敛性
本文研究了用两种非线性数值方法逼近具有测量数据和非均质各向异性扩散矩阵的线性椭圆问题。通过变量u = ψ(v)的变化,两种方法在弱意义上证明了问题连续解的收敛性,其中ψ是(- 1,1)和R之间的一个精选的微分同构,并且v的值在(- 1,1)中。我们首先研究了任意简单网格上的非线性有限元逼近。我们证明了一个离散解的存在性,并在标准正则性条件下,利用Hölder和Sobolev不等式证明了它收敛于问题的弱解。在不属于h1 (Ω)的二维和三维情况下的一些数值结果表明,该方法可以提供准确的结果。然后,我们构造了一个数值格式,该格式在右边属于L1的情况下,对问题的熵弱解具有收敛性。这是通过非线性控制体积有限元(CVFE)方法实现的,保持了相同的非线性重构,并添加了上游加权评估和非线性p -拉普拉斯消失稳定项。
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
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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