Abstract
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.
Highlights
The Theory of Connections (ToC), as introduced in [1], consists of a general analytical framework to obtain constrained expressions, f ( x ), in one-dimension
A constrained expression is a function expressed in terms of another function, g( x ), that is freely chosen and, no matter what the g( x ) is, the resulting expression always satisfies a set of n constraints
This paper extends to n-dimensional spaces the Univariate Theory of Connections (ToC), introduced in Ref. [1]
Summary
The Theory of Connections (ToC), as introduced in [1], consists of a general analytical framework to obtain constrained expressions, f ( x ), in one-dimension. This study first extends the Theory of Connections to two-dimensions by providing, for rectangular domains, all surfaces that are subject to: (1) Dirichlet constraints; (2) Neumann constraints; and (3) any combination of Dirichlet and Neumann constraints This theory is generalized to the Multivariate Theory of Connections which provide in n-dimensional space all possible manifolds that satisfy boundary constraints on the value and boundary constraints on any-order derivative. It shows that the one-dimensional ToC can be used in two dimensions when the constraints (functions or derivatives) are provided along one axis only. This second example is provided to show how to use the general approach given in Equation (1) and described in [1], when derivative constraints are involved.
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