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Abstract
In this article, a technique for developing cubature rules with preassigned nodes is presented to avoid wasting of information in scientific computation. The corresponding constructive method of the cubature rule is also given. As an application of the rules, a cubature formula on disk, which was derived via the method of reproducing kernel in (Xu, Y., 2000, Constructing cubature formulae by the method of reproducing kernel. Numerische Mathematik, 85, 155–173), is reconstructed by using our technique. When the preassigned nodes are selected as the nodes of a cubature formula of lower degree, an embedded cubature formula can be easily obtained by the presented method. Furthermore, some examples are included in the article.
§This paper is dedicated to the 70th birthday of Prof. Zuhair Nashed.
Acknowledgements
Prof. Zuhair Nashed's contributions and devotion to mathematics impacted many people and particularly his students. As one of his student, Fengshan Liu wants to thank Prof. Zuhair Nashed for his academic and personal advise to Fengshan during his PhD study at University of Delaware and during his work at Delaware State University. The project is supported by NNSFC (Nos. 10471018, 60533060), Program of New Century Excellent Fellowship of NECC, and is partially funded by a DoD fund (DAAD19-03-1-0375).
Notes
§This paper is dedicated to the 70th birthday of Prof. Zuhair Nashed.