Adaptive space-time isogeometric analysis for parabolic evolution problems
This work provides a rigorous theoretical foundation and adaptive mesh refinement for space-time IgA methods, benefiting computational scientists solving parabolic PDEs with improved accuracy and efficiency.
The paper develops a locally stabilized space-time isogeometric analysis (IgA) method for parabolic evolution problems, proving coercivity, boundedness, consistency, and asymptotically optimal a priori error estimates. Numerical results confirm improved convergence of global approximation errors and local efficiency of error indicators.
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.