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

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 an efficient a posteriori error bound generation. 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 routines for an auxiliary variables reconstruction, which generate the majorant (or minorant) of an error, can be executed several times faster than the routines for a primal unknown.

A short perspective on a posteriori error control and adaptive discretizations

“…Other exciting directions of research are provided in the following recent contributions including space-time analysis, as well as functional-type error estimates for isogeometric analysis, see Langer et al (2016); Matculevich (2018); Langer et al (2019).…”

5. Adaptive space-time isogeometric analysis for parabolic evolution problems

“…The optimal value for β reads as β := C F m I eq / m I d . According to numerical results obtained in [34,47,43], the most efficient majorant reconstruction is obtained with spline degree q p. At the same time, the approximation u h is reconstructed on the mesh K h , whereas a coarser mesh K Mh , M ∈ N + , is used to recover y h . This helps to minimise the number of d.o.f.…”

“…By exploiting the universality and efficiency of these error estimates as well as taking an advantage of smoothness of the IgA approximations, we aim at constructing fast fully adaptive space-time methods that could tackle complicated problems inspired by industrial applications. These two techniques were already combined in application to elliptic problems in [34] and [47] using tensor-based splines and THB-splines [23,24,22], respectively. Both papers confirmed that the majorants provide not only reliable and efficient upper bounds of the total energy error but a quantitatively sharp indicator of local element-wise errors.…”

Inexpensive polynomial-degree- and number-of-hanging-nodes-robust equilibrated flux a posteriori estimates for isogeometric analysis

“…In the spirit of [Rep99,Rep00a,Rep00b], the works [KT15a, KT15b,Mat18] present guaranteed fully computable upper bounds of the approximation error for tensor-product splines and hierarchical splines, respectively. A second estimate is also derived, giving a lower bound of the error.…”