Guaranteed error control bounds for the stabilised space-time IgA approximations to parabolic problems

TL;DR

This paper develops guaranteed, mesh-independent error bounds for stabilised space-time IgA approximations of parabolic problems, enabling reliable adaptive solvers with proven efficiency.

Contribution

It introduces fully computable, guaranteed error bounds for space-time IgA methods that are independent of mesh parameters and applicable to stabilised approximations.

Findings

Error bounds are reliable and fully computable.

Error indicators improve adaptive mesh refinement.

Numerical examples confirm efficiency and robustness.

Abstract

The paper is concerned with space-time IgA approximations of parabolic initial-boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of IgA approximations and investigate their applicability. The derivation method is based on the analysis of respective integral identities and purely functional arguments. Therefore, the estimates do not contain mesh-dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they provide computable error bounds for norms associated with stabilised space-time IgA approximations as well as imply efficient error indicators enhancing the performance of fully adaptive solvers. The last section of the paper contains a series of numerical examples where approximate solutions are recovered by IgA techniques. The mesh refinement algorithm is governed by a local error indicator generated by the error majorant. Numerical results discussed in the last section illustrate both reliability, as well as the quantitative efficiency of the error estimates presented.