Adaptive space-time isogeometric analysis for parabolic evolution problems

TL;DR

This paper develops a localized space-time isogeometric analysis scheme for parabolic problems, providing theoretical error estimates and an adaptive refinement algorithm, with numerical results demonstrating improved convergence and efficiency.

Contribution

It introduces a localized stabilized space-time IgA scheme with rigorous error analysis and an adaptive mesh refinement method based on functional a posteriori error estimates.

Findings

Asymptotically optimal a priori error estimates achieved.

Numerical results show improved convergence rates.

Error indicators are locally efficient and reliable.

Abstract

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.