  We propose a new class of geometric invariants called jet vanishing orders,
and use them to establish a new selection algorithm in the Kohn's construction
of subelliptic multipliers for special domains in dimension $3$, inspired by
the work of Y.-T. Siu [S10]. In particular, we obtain effective termination of
our selection algorithm with explicit bounds both for the steps of the
algorithm and the order of subellipticity in the corresponding subelliptic
estimates. Our procedure possesses additional features of certain stability
under high order perturbations, due to deferring the step of taking radicals to
the very end of the algorithm.
  We further illustrate by examples the sharpness in our technical results and
demonstrate the complete procedure for arbitrary high order perturbations of
the Catlin-D'Angelo example [CD10] in Section 5.
  Our techniques here may be of broader interest for more general PDE systems,
in the light of the recent program initiated by the breakthrough paper of Y.-T.
Siu [S17].
