### CONSTRUCTION OF CUBATURE FORMULAS VIA BIVARIATE QUADRATIC SPLINE SPACES OVER NON-UNIFORM TYPE-2 TRIANGULATION

Jiang Qian1, Xiquan Shi2, Jinming Wu3, Dianxuan Gong4

1. 1. College of Science, Hohai University, Nanjing 211100, China;
2. Division of Physics, Engineering, Mathematics, and Computer Science, Delaware State University, Dover, DE 19901, USA;
3. Statistics and Mathematics Institute, Zhejiang Gongshang University, Hangzhou 310018, China;
4. College of Science, North China University of Science and Technology, Tangshan 063210, China
• Received:2020-03-30 Revised:2020-06-08 Online:2022-03-15 Published:2022-03-29
• Contact: Jiang Qian,Email: qianjianghhu@sina.com
• Supported by:
This work was supported by the Fundamental Research Funds for the Central Universities of Hohai University (Grant No.2019B19414,2019B44914),the Natural Science Foundation of Jiangsu Province for the Youth (Grant No.BK20160853),Key Laboratory of Ministry of Education for Coastal Disaster and Protection,Hohai University (Grant No.202011),the National Natural Science Foundation of China (Grant No.11601151),and the National Science Foundation of Zhejiang Province (Grant No.LY19A010003).

Jiang Qian, Xiquan Shi, Jinming Wu, Dianxuan Gong. CONSTRUCTION OF CUBATURE FORMULAS VIA BIVARIATE QUADRATIC SPLINE SPACES OVER NON-UNIFORM TYPE-2 TRIANGULATION[J]. Journal of Computational Mathematics, 2022, 40(2): 205-230.

In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S21mn(2)), and coefficients of splines in terms of B-net are calculated 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.

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