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Open AccessFeature PaperArticle

The Multivariate Theory of Connections†

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
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This paper is an extended version of our paper published in Mortari, D. “The Theory of Connections: Connecting Functions.” IAA-AAS-SciTech-072, Forum 2018, Peoples’ Friendship University of Russia, Moscow, Russia, 13–15 November 2018.
Mathematics 2019, 7(3), 296; https://doi.org/10.3390/math7030296
Received: 4 January 2019 / Revised: 25 February 2019 / Accepted: 18 March 2019 / Published: 22 March 2019
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, $g ( x , y )$ , satisfy all constraints no matter what the $g ( x , y )$ is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. View Full-Text
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Mortari, D.; Leake, C. The Multivariate Theory of Connections. Mathematics 2019, 7, 296.

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