A posteriori error estimates for space-time IgA approximations to parabolic initial boundary value problems

TL;DR

This paper develops reliable and efficient a posteriori error estimates for space-time IgA approximations to parabolic initial boundary value problems, enabling adaptive refinement and improved accuracy.

Contribution

It introduces a general functional a posteriori error estimation method for space-time IgA schemes, including error majorants and their equivalence to energy norms.

Findings

Error estimates are mesh-independent and valid for a wide class of IgA approximations.

Introduces flexible error majorants that can be minimized for optimal bounds.

Establishes the efficiency and reliability of the proposed a posteriori error estimates.

Abstract

This work is concerned with a posteriori error estimates of the functional type for approximations constructed by space-time IgA scheme presented in paper by Langer, Neumueller, and Moore (2016). We consider approxima- tions in the corresponding IgA spaces based on elliptic and bounded bilinear form (associated with the spatial part). It is proved that the approximations satisfy classic a priori error estimates. Also, we deduce a posteriori error estimates for a stabilized weak formulation of the considered parabolic initial boundary value problem (I-BVP). They are derived by a general functional method and do not contain mesh dependent constants. The estimates are valid for a wide class of approximations. In particular, they imply estimates for the discrete norm of IgA approximations. Moreover, we introduce different forms of a posteriori error estimates (error majorants) and establish equivalence of majorants and energy error norm. This property justifies efficiency and reliability of a posteriori error estimates. Another important property of the estimates is their flexibility with respect to a certain amount of free parameters. Using these parameters we can obtain estimates for different error norms and minimize the respective majorant in order to find the best possible bound of the error.