# 超収束

## 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${\displaystyle L^{2}}$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．

## 出典

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2. ^ 室谷義昭, 石渡恵美子, & Brunner, H. 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, 京都大学数理解析研究所講究録 1396,(2004) 72-84.
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4. ^ Yamamoto, T., Fang, Q., & Chen, X. (2001). Superconvergence and nonsuperconvergence of the Shortley-Weller approximations for Dirichlet problems. Numerical Functional Analysis and Optimization, 22(3-4), 455-470.
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, B., & Fu, G. (2017). Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements. ESAIM: Mathematical Modelling and Numerical Analysis, 51(1), 365-398.
14. ^ 橘基. (1997). 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, 96(2), B28-B30.
15. ^ 坂下秀男. (1971). 収束円上の超収束について (Notes on the over-convergence on the circle of convergence). 京都教育大学紀要. B, 自然科学, 39, 1-6.
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.