Volume 40, Issue 2
Construction of Cubature Formulas via Bivariate Quadratic Spline Spaces over Non-Uniform Type-2 Triangulation

J. Comp. Math., 40 (2022), pp. 205-230.

Published online: 2022-01

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• Abstract

In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in $S^1_2 (∆^{(2)}_{mn})$, and coefficients of splines in terms of B-net are reviewed firstly. Moreover, by means of coefficients in terms of B-net, computation of bivariate numerical cubature over triangular sub-domains with respect to variables $x$ and $y$ is transferred into summation of coefficients of splines in terms of B-net. Thus concise bivariate cubature formulas are constructed over rectangular sub-domain. Furthermore, by means of module of continuity and max-norms, error estimates for cubature formulas are derived over both sub-domains and the domain.

• Keywords

Multivariate spline, Bivariate cubature, Conformality of Smoothing Cofactor Method, B-net, Non-uniform Type-2 Triangulation.

65D07, 65D32

xqshi2000@gmail.com (Xiquan Shi)

qianjianghhu@sina.com (Jiang Qian)

wujm97@zjgsu.edu.cn (Jinming Wu)

dxgong@heut.edu.cn (Dianxuan Gong)

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@Article{JCM-40-205, author = {Xiquan and Shi and xqshi2000@gmail.com and 21993 and Division of Physics, Engineering, Mathematics, and Computer Science, Delaware State University, Dover, DE 19901, U.S.A and Xiquan Shi and Jiang and Qian and qianjianghhu@sina.com and 21992 and College of Science, Hohai University, Nanjing 211100, China and Jiang Qian and Jinming and Wu and wujm97@zjgsu.edu.cn and 21994 and Statistics and Mathematics Institute, Zhejiang Gongshang University, Hangzhou 310018, China and Jinming Wu and Dianxuan and Gong and dxgong@heut.edu.cn and 21995 and College of Science, North China University of Science and Technology, Tangshan 063210, China and Dianxuan Gong}, title = {Construction of Cubature Formulas via Bivariate Quadratic Spline Spaces over Non-Uniform Type-2 Triangulation}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {205--230}, abstract = {

In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in $S^1_2 (∆^{(2)}_{mn})$, and coefficients of splines in terms of B-net are reviewed firstly. Moreover, by means of coefficients in terms of B-net, computation of bivariate numerical cubature over triangular sub-domains with respect to variables $x$ and $y$ is transferred into summation of coefficients of splines in terms of B-net. Thus concise bivariate cubature formulas are constructed over rectangular sub-domain. Furthermore, by means of module of continuity and max-norms, error estimates for cubature formulas are derived over both sub-domains and the domain.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2008-m2020-0077}, url = {http://global-sci.org/intro/article_detail/jcm/20184.html} }
TY - JOUR T1 - Construction of Cubature Formulas via Bivariate Quadratic Spline Spaces over Non-Uniform Type-2 Triangulation AU - Shi , Xiquan AU - Qian , Jiang AU - Wu , Jinming AU - Gong , Dianxuan JO - Journal of Computational Mathematics VL - 2 SP - 205 EP - 230 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2008-m2020-0077 UR - https://global-sci.org/intro/article_detail/jcm/20184.html KW - Multivariate spline, Bivariate cubature, Conformality of Smoothing Cofactor Method, B-net, Non-uniform Type-2 Triangulation. AB -

In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in $S^1_2 (∆^{(2)}_{mn})$, and coefficients of splines in terms of B-net are reviewed firstly. Moreover, by means of coefficients in terms of B-net, computation of bivariate numerical cubature over triangular sub-domains with respect to variables $x$ and $y$ is transferred into summation of coefficients of splines in terms of B-net. Thus concise bivariate cubature formulas are constructed over rectangular sub-domain. Furthermore, by means of module of continuity and max-norms, error estimates for cubature formulas are derived over both sub-domains and the domain.

Jiang Qian, Xiquan Shi, Jinming Wu & Dianxuan Gong. (2022). Construction of Cubature Formulas via Bivariate Quadratic Spline Spaces over Non-Uniform Type-2 Triangulation. Journal of Computational Mathematics. 40 (2). 205-230. doi:10.4208/jcm.2008-m2020-0077
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