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.
Parallel matching-based AMG preconditioners for elliptic equations discretized by IgA
National Research Council of Italy (CNR)（意大利国家研究委员会）；Institute for Applied Computing “Mauro Picone” (IAC)（毛罗·皮切纳应用计算研究所）；University of Pisa（比萨大学）；University of Rome “Tor Vergata”（罗马托维尔加塔大学）
Space-Time Isogeometric Analysis of Parabolic Evolution Equations
Austrian Academy of Sciences（奥地利科学院）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉德纳计算与应用数学研究所）；Institute of Computational Mathematics, Johannes Kepler University（约翰内斯·开普勒大学计算数学研究所）
Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM
University of Parma（帕尔马大学）；University of Cassino and southern Lazio（卡西诺及南拉齐奥大学）；University of Siena（锡耶纳大学）；University of Pavia（帕维亚大学）；University of Florence（佛罗伦萨大学）
Discontinuous Galerkin Isogeometric Analysis on Non-matching Segmentation: Error Estimates and Efficient Solvers
Johannes Kepler University (JKU)（约翰内斯·开普勒大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院计算与应用数学约翰·拉德恩研究所）
State Key Laboratory of Scientific；Engineering Computing Academy of Mathematics；Systems Science Chinese Academy of Sciences P.O. Box 2719, Beijing 100190, CHINA
Adaptive refinement in defeaturing problems via an equilibrated flux a posteriori error estimator
Institute of Mathematics, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院数学研究所）；IMATI CNR（CNR IMATI研究所）；MOX, Department of Mathematics, Politecnico di Milano（米兰理工大学数学系MOX实验室）；GNCS INdAM Group（INdAM GNCS小组）；Departamento de Matemática Aplicada, Universidade de Santiago de Compostela（圣地亚哥德孔波斯特拉大学应用数学系）；Galician Centre for Mathematical Research and Technology (CITMAga)（加利西亚数学研究与技术中心）
Adaptive finite element method for elliptic optimal control problems: convergence and optimality
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院系统科学研究所）
Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes
School of Mathematical Science, Tongji University（同济大学数学科学学院）；Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education（同济大学教育部智能计算与应用重点实验室）
A correction adaptive two-grid finite element method for nonselfadjoint or indefinite elliptic problems
Lingnan Normal University（岭南师范学院）；Missouri University of Science and Technology（密苏里科技大学）；Guangzhou University（广州大学）；South China Normal University（华南师范大学）
