{"title":"一般二阶椭圆 PDE 的 AFEM 的准光学性","authors":"Arnab Pal, Thirupathi Gudi","doi":"10.1515/cmam-2023-0238","DOIUrl":null,"url":null,"abstract":"In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0238_eq_0388.png\" /> <jats:tex-math>{b\\in[L^{\\infty}(\\Omega)]^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0238_eq_0235.png\" /> <jats:tex-math>{H^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. 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引用次数: 0
摘要
本文针对对流项 b∈ [ L ∞ ( Ω ) ] d {b\in[L^{\infty}(\Omega)]^{d}} 的一般二阶非自洽椭圆 PDE，并利用对偶问题的最小正则性（即、对偶问题的解只有 H 1 {H^{1}} -正则性，从而扩展了结果[J.M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer.Anal.46 2008, 5, 2524-2550].数值实验说明了理论结果。
Quasi-Optimality of an AFEM for General Second Order Elliptic PDE
In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b∈[L∞(Ω)]d{b\in[L^{\infty}(\Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H1{H^{1}}-regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.
期刊介绍：
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.
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