  The 3D incompressible Euler equation is an important research topic in the
mathematical study of fluid dynamics. Not only is the global regularity for
smooth initial data an open issue, but the behaviour may also depend on the
presence or absence of boundaries.
  For a good understanding, it is crucial to carry out, besides mathematical
studies, high-accuracy and well-resolved numerical exploration. Such studies
can be very demanding in computational resources, but recently it has been
shown that very substantial gains can be achieved first, by using Cauchy's
Lagrangian formulation of the Euler equations and second, by taking advantages
of analyticity results of the Lagrangian trajectories for flows whose initial
vorticity is H\"older-continuous. The latter has been known for about twenty
years (Serfati, 1995), but the combination of the two, which makes use of
recursion relations among time-Taylor coefficients to obtain constructively the
time-Taylor series of the Lagrangian map, has been achieved only recently
(Frisch and Zheligovsky, 2014; Podvigina {\em et al.}, 2016 and references
therein).
  Here we extend this methodology to incompressible Euler flow in an
impermeable bounded domain whose boundary may be either analytic or have a
regularity between indefinite differentiability and analyticity.
Non-constructive regularity results for these cases have already been obtained
by Glass {\em et al.} (2012). Using the invariance of the boundary under the
Lagrangian flow, we establish novel recursion relations that include
contributions from the boundary. This leads to a constructive proof of
time-analyticity of the Lagrangian trajectories with analytic boundaries, which
can then be used subsequently for the design of a very high-order
Cauchy--Lagrangian method.
