 
Summary: Trivariate Spline Approximation of
DivergenceFree Vector Fields
Gerard Awanou 1) and MingJun Lai 2)
Abstract. We discuss the approximation properties of divergencefree vector
elds by using trivariate spline vectors which are also divergencefree. 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 maxnorm
and L p norm.
x1. Introduction
In this paper we show how trivariate spline vectors which are divergencefree can
approximate any given divergencefree vector eld. More precisely, we give appro
ximation properties of these spline spaces and track the approximation constants.
Throughout the paper, we will assume
that
is a bounded, simplyconnected 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) dierentiable on
i = 1; 2; 3. Recall that f is a divergencefree vector if
