非均匀二型三角剖分二元二次样条的数值积分公式  

Numerical integration formulas of bivariate quadratic splines upon non-uniform type-2 triangulation

作　　者：钱江[1,2] 王凡 QIAN Jiang;WANG Fan(College of Science,Hohai University,Nanjing 211100,China;Key Laboratory of Coastal Disaster and Defence(Hohai University),Ministry of Education,Nanjing 210098,China;College of Science,Nanjing Agricultural University,Nanjing 210095,China)

机构地区：[1]河海大学理学院,江苏南京211100 [2]海岸灾害及防护教育部重点实验室(河海大学),江苏南京210098 [3]南京农业大学理学院,江苏南京210095

出　　处：《浙江大学学报（理学版）》2022年第5期555-563,579,共10页Journal of Zhejiang University（Science Edition）

基　　金：江苏省自然科学基金青年基金项目(BK20160853);河海大学中央高校基本科研业务费项目(2019B19414);海岸灾害与防护教育部重点实验室开放基金项目(河海大学202011);国家留学基金资助出国留学项目(访问学者201806715010)。

摘　　要：给出了构成矩形域的4个三角形子区域的二元样条拟插值算子的等价形式,对这4个三角形子区域分别建立了数值积分公式,相加后得到一般矩形域上的数值积分公式,同时给出了构造数值积分公式所需的结点处的函数值与相应的求积系数。进一步,利用算子范数、连续模及拟插值算子的保多项式性,针对具有不同连续性的被积函数,得到了相应的数值积分公式的求积余项。研究表明,提出的数值积分公式不仅具有较高的计算精度,而且计算量约为二元张量积型求积公式的1/5。数值算例进一步说明了数值积分公式的有效性。In the paper, equivalent representations of the spline quasi-interpolation are presented over four triangular sub-domains contained in a general rectangular domain. Moreover, the direct bivariate numerical integration formulas are constructed over the four triangular sub-domains, while summing them yields the integration formula over the general rectangular domain. For illustration, the necessary function values and the corresponding integration coefficients are listed in several tables. Furthermore, based on the norm of the operator, the module of continuity and the reproduction of bivariate polynomials, error estimations of the numerical integration are derived for continuously differential functions with different orders. The computational cost of the proposed method is approximately 1/5 of that based on tensor-product-type quadratic splines. Numerical examples show the validity of the proposed numerical integration approach.

关 键 词：多元样条 光滑余因子协调法 二元数值积分 样条拟插值 B网 

分 类 号：O241.5[理学—计算数学]