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Trivariate Spline Approximation of Divergence-Free Vector Fields

Summary: Trivariate Spline Approximation of
Divergence-Free Vector Fields
Gerard Awanou 1) and Ming-Jun Lai 2)
Abstract. We discuss the approximation properties of divergence-free vector
elds by using trivariate spline vectors which are also divergence-free. We pay
special attention to the approximation constants and show that they depend
only on the smallest solid angle in the underlying tetrahedral partition and the
nature of the boundary of the domain. The estimates are given in the max-norm
and L p norm.
x1. Introduction
In this paper we show how trivariate spline vectors which are divergence-free can
approximate any given divergence-free vector eld. More precisely, we give appro-
ximation properties of these spline spaces and track the approximation constants.
Throughout the paper, we will assume
is a bounded, simply-connected do-
main of IR 3 with a boundary of class C m;1 ; m  0.
Let f = (f 1 ; f 2 ; f 3 ) be a vector with components f i (x; y; z) di erentiable on

i = 1; 2; 3. Recall that f is a divergence-free vector if


Source: Awanou, Gerard - Department of Mathematical Sciences, Northern Illinois University


Collections: Mathematics