Guaranteed error bounds and local indicators for adaptive solvers using stabilised space-time IgA approximations to parabolic problems

Abstract

The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and investigate their efficiency. The derivation of error estimates is based on the analysis of the corresponding integral identity and exploits purely functional arguments in the maximal parabolic regularity setting. The estimates are valid for any approximation from the admissible (energy) class and do not contain mesh-dependent constants. They provide computable and fully guaranteed error bounds for the norms arising in stabilised space–time approximations. Furthermore, a posterior error estimates yield efficient error indicators enhancing the performance of adaptive solvers and generate very successful mesh refinement procedures. Theoretical results are verified with a series of numerical examples, in which approximate solutions and the corresponding fluxes are recovered by IgA techniques. The numerical results confirm the high efficiency of the method in the context of the two main goals of a posteriori error analysis: estimation of global errors and mesh adaptation.