  In this paper we present a model to calculate the stringy product on twisted
orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds.
  In the first part we consider the non-twisted case on an orbifold presented
as the quotient of a manifold acted by a compact abelian Lie group. We give an
explicit description of the obstruction bundle, we explain the relation with
the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map,
with the Chen-Ruan cohomology, and we explicitely calculate the stringy product
for a weighted projective orbifold.
  In the second part we consider orbifolds presented as the quotient of a
manifold acted by a finite abelian group and twistings coming from the group
cohomology. We show a decomposition formula for twisted orbifold K-theory that
is suited to calculate the stringy product and we use this formula to calculate
two examples when the group is $(\integer/2)^3$.
