Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

TL;DR

This paper investigates functional a posteriori error estimates for IgA schemes in elliptic boundary-value problems, demonstrating their efficiency and sharper control compared to residual-based estimates through numerical testing.

Contribution

It introduces a novel functional error estimation approach using B-(THB-)splines for auxiliary variable reconstruction, enabling coarser meshes and faster computations.

Findings

Functional error estimates are sharper than residual-based estimates.

The auxiliary variable routines are computationally efficient.

Numerical results confirm the effectiveness of the proposed error bounds.

Abstract

This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundary-value problems. Along with the detailed discussion of the most crucial properties of such estimates, we present the algorithm of a reliable solution approximation together with the scheme of efficient a posteriori error bound generation-based on solving an auxiliary problem with respect to an introduced vector-valued variable. In this approach, we take advantage of B-(THB-)spline's high smoothness for the auxiliary vector function reconstruction, which, at the same time, allows to use much coarser meshes and decrease the number of unknowns substantially. The most representative numerical results, obtained during a systematic testing of error estimates, are presented in the second part of the paper. The efficiency of the obtained error bounds is analysed from both the error estimation (indication) and the computational expenses points of view. Several examples illustrate that functional error estimates (alternatively referred to as the majorants and minorants of deviation from an exact solution) perform a much sharper error control than, for instance, residual-based error estimates. Simultaneously, assembling and solving the routines for an auxiliary variable reconstruction which generate the majorant of an error can be executed several times faster than the routines for a primal unknown.