Adaptive space-time isogeometric analysis for parabolic evolution problems

The paper proposes new locally stabilized space-time isogeometric analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) have been presented and studied by U. Langer, M. Neumuller, and S. Moore (2016). The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time isogeometric analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that have been introduced by S. Repin (2002), and later rigorously studied in the context of isogeometric analysis by U. Langer, S. Matculevich, and S. Repin (2017). Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.