超収束のソースを表示

[[数値解析]]において'''超収束''' （ちょうしゅうそく、Superconvergence） とは、[[常微分方程式の数値解法]]・[[偏微分方程式の数値解法]]において通常より収束が早くなる現象をさす。このような現象は[[有限要素法]]<ref>Kříek, M. (1994). Superconvergence phenomena in the finite element method. Computer methods in applied mechanics and engineering, 116(1-4), 157-163.</ref>・[[選点法]]<ref>{{Cite journal|和書|author=室谷義昭, 石渡恵美子, Brunner Hermann |date=2004-10 |url=https://hdl.handle.net/2433/25969 |title=比例的遅れを持つ積分及び微分方程式に対する選点法の超収束について (情報科学と函数解析の接点 : これまでとこれから) |journal=数理解析研究所講究録 |ISSN=1880-2818 |publisher=京都大学数理解析研究所 |volume=1396 |pages=72-84 |hdl=2433/25969 |CRID=1050564285465699072}}</ref>やShortley-Weller近似 （[[差分法]]の一つ）<ref>Matsunaga, N., & Yamamoto, T. (2000). Superconvergence of the Shortley–Weller approximation for Dirichlet problems. [[:en:Journal of computational and applied mathematics]], 116(2), 263-273.</ref><ref>{{Cite journal |author=Yamamoto, Tetsuro; Fang, Qing; Chen, Xiaojun |year=2001 |title=Superconvergence and nonsuperconvergence of the Shortley-Weller approximations for Dirichlet problems |url=https://doi.org/10.1081/NFA-100105113 |journal=Numerical Functional Analysis and Optimization |volume=22 |issue=3-4 |pages=455-470 |doi=10.1081/NFA-100105113 |publisher=Taylor & Francis}}</ref><ref>Li, Z. C., Fang, Q., & Wang, S. (2009). Superconvergence of solution derivatives for the Shortley–Weller difference approximation for parabolic problems. Numerical Functional Analysis and Optimization, 30(11-12), 1360-1380.</ref><ref>Yoon, G., & Min, C. (2014). Supra-convergences of Shortley–Weller method for Poisson equation. Appl. Numer. Math.</ref><ref>Weynans, L. (2015). A proof in the finite-difference spirit of the superconvergence of the gradient for the Shortley-Weller method (Doctoral dissertation, INRIA Bordeaux; [[INRIA]]).</ref>などで見られる。

==HDG法の超収束について==
hybrid 不連続 Galerkin（HDG）法<ref>Oikawa, I., & Kikuchi, F. (2010). Discontinuous Galerkin FEM of hybrid type. JSIAM Letters, 2, 49-52.</ref>（[[:en:discontinuous Galerkin method|不連続 Galerkin法]]<ref>Reed, W. H., & Hill, T. R. (1973). Triangular mesh methods for the neutron transport equation (No. LA-UR-73-479; CONF-730414-2). Los Alamos Scientific Lab., N. Mex.(USA).</ref>の改良）の超収束性に関して研究が進展し、様々な結果が得られている。それらは大きく分けて、数値流束の安定化項に<math>L^2</math>射影を施すLehrenfeld-Schöberl安定化<ref>C. Lehrenfeld. Hybrid discontinuous Galerkin methods for solving incompressible flow problems. RheinischWestfalischen Technischen Hochschule Aachen, 2010.</ref>と、 HDG射影を用いるM-decomposition理論<ref>Cockburn, B., Fu, G., & Sayas, F. (2017). Superconvergence by 𝑀-decompositions. Part I: General theory for HDG methods for diffusion. Mathematics of Computation, 86(306), 1609-1641.</ref><ref>Cockburn, B., & Fu, G. (2017). Superconvergence by M-decompositions. Part II: Construction of two-dimensional finite elements. ESAIM: Mathematical Modelling and Numerical Analysis, 51(1), 165-186.</ref><ref>{{Cite journal |author=Cockburn, Bernardo; Fu, Guosheng |year=2017 |title=Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements |url=https://doi.org/10.1051/m2an/2016023 |journal=ESAIM: Mathematical Modelling and Numerical Analysis |volume=51 |issue=1 |pages=365-398 |doi=10.1051/m2an/2016023 |publisher=EDP Sciences}}</ref>との２つに分類される。

==他の分野における超収束==
* [[ゲージ理論]]における超収束<ref>{{Cite journal|和書|author=橘基 |year=1997 |url=https://doi.org/10.24532/soken.96.2_b28 |title=超対称ゲージ理論における超収束関係式(場及び弦の量子論における非摂動的手法,研究会報告) |journal=素粒子論研究 |ISSN=03711838 |publisher=素粒子論グループ 素粒子論研究 編集部 |volume=96 |issue=2 |pages=B28-B30 |doi=10.24532/soken.96.2_b28 |CRID=1390282679054028928}}</ref>
* [[:en:approximation theory|近似理論]]における超収束 （{{lang-en-short|Overconvergence}}）<ref> {{Cite journal |author=坂下秀男 |date=1971-09 |url=https://hdl.handle.net/20.500.12176/2945 |title=Notes on the over-convergence on the circle of convergence |trans-title=抄訳:収束円上の超収束について |journal=京都教育大学紀要.B, 自然科学 |ISSN=0023-6101 |publisher=京都教育大学 |volume=39 |pages=1-6 |hdl=20.500.12176/2945 |id={{CRID|1050845762911800320}}}}</ref><ref>Gal, S. G. (2013). Overconvergence in Complex Approximation. New York: Springer.</ref><ref>Gal, S. G. (2009). Approximation by complex Bernstein and convolution type operators. Singapore: [[:en:World scientific]].</ref><ref>Chui, C. K., & Parnes, M. N. (1971). Approximation by overconvergence of a power series. [[:en:Journal of Mathematical Analysis and Applications]], 36(3), 693-696.</ref><ref>Walsh, J. L., Interpolation and approximation by rational functions in the complex domain. [[アメリカ数学会|American Mathematical Society]].</ref>

==関連項目==
*[[数列の加速法]] （超収束の[[数列]]版）

==出典==
{{Reflist}}

==参考文献==
* {{Citation | last1=Barbeiro | first1=S. | last2=Ferreira | first2=J.&nbsp;A. | last3=Grigorieff | first3=R.&nbsp;D. | title=Supraconvergence of a finite difference scheme for solutions in ''H''<sup>''s''</sup>(0, ''L'') | journal=IMA J Numer Anal | volume=25 | issue=4 | pages=797–811 | year=2005 | doi=10.1093/imanum/dri018 | url=https://doi.org/10.1093/imanum/dri018}}
* {{Citation | last1=Ferreira | first1=J.&nbsp;A. | last2=Grigorieff | first2=R.&nbsp;D. | title=On the supraconvergence of elliptic finite difference methods | journal=Applied Numerical Mathematics | volume=28 | issue=2-4 | pages=275–292 | year=1998 |doi=10.1016/S0168-9274(98)00048-8 |url=https://doi.org/10.1016/S0168-9274(98)00048-8 |publisher=Elsevier}}
* {{Citation | last1=Levine | first1=N.&nbsp;D. | title=Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations | journal=IMA J Numer Anal | volume=5 | issue= | pages=407–427 | year=1985 | doi=10.1093/imanum/5.4.407 | url=https://doi.org/10.1093/imanum/5.4.407}}

{{偏微分方程式の数値解法}}

{{デフォルトソート:ちようしゆうそく}}
[[Category:収束]]
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