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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^[1]・選点法^[2]やShortley-Weller近似（差分法の一つ）^[3]^[4]^[5]^[6]^[7]などで見られる。

HDG法の超収束について

hybrid 不連続 Galerkin（HDG）法^[8]（不連続 Galerkin法^[9]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 $L^{2}$ 射影を施すLehrenfeld-Schöberl安定化^[10]と、 HDG射影を用いるM-decomposition理論^[11]^[12]^[13]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^[14]
近似理論における超収束（英: Overconvergence）^[15]^[16]^[17]^[18]^[19]

出典

^ Kříek, M. (1994). Superconvergence phenomena in the finite element method. Computer methods in applied mechanics and engineering, 116(1-4), 157-163.
^ 室谷義昭, 石渡恵美子, Brunner Hermann「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)」『数理解析研究所講究録』第1396巻、京都大学数理解析研究所、2004年10月、72-84頁、CRID 1050564285465699072、hdl:2433/25969、ISSN 1880-2818。
^ 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.
^ Yamamoto, Tetsuro; Fang, Qing; Chen, Xiaojun (2001). “Superconvergence and nonsuperconvergence of the Shortley-Weller approximations for Dirichlet problems”. Numerical Functional Analysis and Optimization (Taylor & Francis) 22 (3-4): 455-470. doi:10.1081/NFA-100105113. https://doi.org/10.1081/NFA-100105113.
^ 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.
^ Yoon, G., & Min, C. (2014). Supra-convergences of Shortley–Weller method for Poisson equation. Appl. Numer. Math.
^ 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).
^ Oikawa, I., & Kikuchi, F. (2010). Discontinuous Galerkin FEM of hybrid type. JSIAM Letters, 2, 49-52.
^ 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).
^ C. Lehrenfeld. Hybrid discontinuous Galerkin methods for solving incompressible flow problems. RheinischWestfalischen Technischen Hochschule Aachen, 2010.
^ 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.
^ 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.
^ Cockburn, Bernardo; Fu, Guosheng (2017). “Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements”. ESAIM: Mathematical Modelling and Numerical Analysis (EDP Sciences) 51 (1): 365-398. doi:10.1051/m2an/2016023. https://doi.org/10.1051/m2an/2016023.
^ 橘基「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)」『素粒子論研究』第96巻第2号、素粒子論グループ素粒子論研究編集部、1997年、B28-B30、CRID 1390282679054028928、doi:10.24532/soken.96.2_b28、ISSN 03711838。
^ 坂下秀男 (1971-09). “Notes on the over-convergence on the circle of convergence [抄訳:収束円上の超収束について]”. 京都教育大学紀要.B, 自然科学 (京都教育大学) 39: 1-6. hdl:20.500.12176/2945. ISSN 0023-6101. CRID 1050845762911800320.
^ Gal, S. G. (2013). Overconvergence in Complex Approximation. New York: Springer.
^ Gal, S. G. (2009). Approximation by complex Bernstein and convolution type operators. Singapore: en:World scientific.
^ 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.
^ Walsh, J. L., Interpolation and approximation by rational functions in the complex domain. American Mathematical Society.

参考文献

Barbeiro, S.; Ferreira, J. A.; Grigorieff, R. D. (2005), “Supraconvergence of a finite difference scheme for solutions in H^s(0, L)”, IMA J Numer Anal 25 (4): 797–811, doi:10.1093/imanum/dri018, https://doi.org/10.1093/imanum/dri018
Ferreira, J. A.; Grigorieff, R. D. (1998), “On the supraconvergence of elliptic finite difference methods”, Applied Numerical Mathematics (Elsevier) 28 (2-4): 275–292, doi:10.1016/S0168-9274(98)00048-8
Levine, N. D. (1985), “Superconvergent Recovery of the Gradient from Piecewise Linear Finite-element Approximations”, IMA J Numer Anal 5: 407–427, doi:10.1093/imanum/5.4.407

[1] Kříek, M. (1994). Superconvergence phenomena in the finite element method. Computer methods in applied mechanics and engineering, 116(1-4), 157-163.

[2] 室谷義昭, 石渡恵美子, Brunner Hermann「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)」『数理解析研究所講究録』第1396巻、京都大学数理解析研究所、2004年10月、72-84頁、CRID 1050564285465699072、hdl:2433/25969、ISSN 1880-2818。

[3] 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.

[4] Yamamoto, Tetsuro; Fang, Qing; Chen, Xiaojun (2001). “Superconvergence and nonsuperconvergence of the Shortley-Weller approximations for Dirichlet problems”. Numerical Functional Analysis and Optimization (Taylor & Francis) 22 (3-4): 455-470. doi:10.1081/NFA-100105113. https://doi.org/10.1081/NFA-100105113.

[5] 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.

[6] Yoon, G., & Min, C. (2014). Supra-convergences of Shortley–Weller method for Poisson equation. Appl. Numer. Math.

[7] 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).

[8] Oikawa, I., & Kikuchi, F. (2010). Discontinuous Galerkin FEM of hybrid type. JSIAM Letters, 2, 49-52.

[9] 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).

[10] C. Lehrenfeld. Hybrid discontinuous Galerkin methods for solving incompressible flow problems. RheinischWestfalischen Technischen Hochschule Aachen, 2010.

[11] 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.

[12] 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.

[13] Cockburn, Bernardo; Fu, Guosheng (2017). “Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements”. ESAIM: Mathematical Modelling and Numerical Analysis (EDP Sciences) 51 (1): 365-398. doi:10.1051/m2an/2016023. https://doi.org/10.1051/m2an/2016023.

[14] 橘基「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)」『素粒子論研究』第96巻第2号、素粒子論グループ素粒子論研究編集部、1997年、B28-B30、CRID 1390282679054028928、doi:10.24532/soken.96.2_b28、ISSN 03711838。

[15] 坂下秀男 (1971-09). “Notes on the over-convergence on the circle of convergence [抄訳:収束円上の超収束について]”. 京都教育大学紀要.B, 自然科学 (京都教育大学) 39: 1-6. hdl:20.500.12176/2945. ISSN 0023-6101. CRID 1050845762911800320.

[16] Gal, S. G. (2013). Overconvergence in Complex Approximation. New York: Springer.

[17] Gal, S. G. (2009). Approximation by complex Bernstein and convolution type operators. Singapore: en:World scientific.

[18] 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.

[19] Walsh, J. L., Interpolation and approximation by rational functions in the complex domain. American Mathematical Society.

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超収束

HDG法の超収束について

他の分野における超収束

関連項目

出典

参考文献