  The paper is concerned with locally stabilized space-time IgA approximations
to initial boundary value problems of the parabolic type. Originally, similar
schemes (but weighted with a global mesh parameter) was presented and studied
by U. Langer, M. Neumueller, and S. Moore (2016). The current work devises a
localised version of this scheme and establishes coercivity, boundedness, and
consistency of the corresponding bilinear form. Using these fundamental
properties together with the corresponding approximation error estimates for
B-splines, we show that the space-time IgA solutions generated by the new
scheme satisfy asymptotically optimal a priori discretization error estimates.
The adaptive mesh refinement algorithm proposed in the paper is based on a
posteriori error estimates of the functional type that has been rigorously
studied in earlier works by S. Repin (2002) and U. Langer, S. Matculevich, and
S. Repin (2017). Numerical results presented in the second part of the paper
confirm the improved convergence of global approximation errors. Moreover,
these results also confirm the local efficiency of the error indicators
produced by the error majorants.
