Error Control, Adaptive Discretizations, and Applications, Part 1

1st Edition, Volume 58 - June 28, 2024
Editors: Franz Chouly, Stéphane P.A. Bordas, Roland becker, Pascal Omnes
Language: English
Hardback ISBN:
9 7 8 - 0 - 4 4 3 - 2 9 4 4 8 - 8
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 9 4 4 9 - 5

Error Control, Adaptive Discretizations, and Applications, Volume 58, Part One highlights new advances in the field, with this new volume presenting interesting chapters writte… Read more

Error Control, Adaptive Discretizations, and Applications, Part 1

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Error Control, Adaptive Discretizations, and Applications, Volume 58, Part One highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this release cover hp adaptive Discontinuous Galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems,
An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements, and Model reduction techniques for parametrized nonlinear partial differential equations.

Cover image
Title page
Table of Contents
Copyright
Contributors
Preface
Chapter One: A short perspective on a posteriori error control and adaptive discretizations
Abstract
1 Introduction
2 A few basic notions
3 Theory of adaptive finite element methods
4 Current engineering practice(s)
5 Special topics and industrial applications
6 Perspectives
7 Chapters
References
Chapter Two: hp adaptive Discontinuous Galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems
Abstract
1 Introduction
2 Governing equations and turbulence modeling
3 Numerical methodology
4 A posteriori error estimation
5 Adaptation strategy
6 Validation
7 Applications
8 Outcomes and perspectives
References
Chapter Three: An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements
Abstract
1 Introduction
2 Notation and classical error estimates
3 Definition of an optimal mesh
4 Construction of an enriched solution and error estimation
5 Local operations on the mesh
6 Academic examples
7 Applications
8 Conclusions
Acknowledgments
References
Chapter Four: Model reduction techniques for parametrized nonlinear partial differential equations
Abstract
1 Introduction
2 Hyper-reduction methods
3 First-order empirical interpolation method
4 Projection-based model reduction methods
5 Nonintrusive model reduction techniques
6 Concluding remarks
Acknowledgement
References
Chapter Five: Adaptive finite elements for obstacle problems
Abstract
1 Introduction
2 Applications
3 Variational formulation
4 Computational demonstrations
5 Discussion
References
Chapter Six: A posteriori error identities and estimates of modelling errors
Abstract
1 Introduction
2 Mathematical background
3 Error identities
4 Simplification of coefficients
5 Errors stipulated by geometric simplifications
6 Dimension reduction errors
References
Chapter Seven: Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction
Abstract
1 Introduction
2 Preliminaries
3 Exact a posteriori error estimation for convex minimization problems
4 Model problems
5 Equivalence to residual type error estimators
6 Quasi-optimality of node-averaging operator
7 Numerical experiments
8 Conclusion
References
Chapter Eight: Algebraic error in numerical PDEs and its estimation
Abstract
1 Introduction
2 Model problem and notation
3 Errors of different origin (and different behavior)
4 Residual-based error estimators and adaptive mesh refinement
5 Error estimates based on flux reconstructions—all-in-one
6 Conclusions
Acknowledgment
References

Franz Chouly

Franz Chouly is professor at the University of Burgundy (Dijon, France). His research is focused on the numerical analysis of partial differential equations and variational inequalities, finite element methods, and applications in fluid and solid mechanics.

Affiliations and expertise

Professor, University of Burgundy, Dijon, France

Stéphane P.A. Bordas

Stéphane is a multi-disciplinary computational and data science researcher, educator, mentor and coach. He was trained as an engineer and applied mathematician who has been teaching and researching in computational sciences since year 1999, in various capacities. He has been in the top 1% most cited in his field, worldwide since year 2015 (ISI Clarivate). Stéphane leads the Legato Team (legato-team.eu), a multi-disciplinary team of about 30 researchers of a dozen nationalities. He is focusing on bringing the rigour of mathematics to bring intuition into the behaviour of complex systems. In particular, he pioneered new approaches to guarantee the quality of surgical simulation devices. The philosophy that he has been following is to create methodologies which translate across discipline boundaries. For example, the methodological backbone of his PhD thesis supports applications in fracture mechanics, nanoscale heterogeneities, biofilm growth, cancer growth, astrocytic metabolism and many others. Recently, his team has become involved, through the Institute of Advanced Studies of the University of Luxembourg in the nascent field of Computational Archaeology. Currently, one of the main focus points of his Team is to bring machine learning tools to bear on mathematical models of physical phenomena. In particular, his group develops adaptive data assimilation, model selection and discretisation optimisation schemes for the deformation of soft matter under large deformation with applications to surgical simulations and robotics. His team has been applying such ideas to programmable matter, multi-scale material modelling, wind energy harvesting, chemical engineering process optimisation, among others. Stéphane has taught over 5,000 students directly and given short courses and research seminars reaching thousands of attendees. He has extensive experience in one-to-one tutoring, mentoring and coaching across various disciplines. He has directly worked with over four hundred collaborators and over fifty different companies, worldwide, as an R&D consultant. Stéphane and his students and collaborators received multiple international prizes for their research and mentorship. He has raised over 28 million euros in research funding from the private and public sector alike. He is Fellow of the Learned Society of Wales, and recipient of the 2022 Eugenio Beltrami Senior Scientist Prize. He is Editor in Chief of Advances in Applied Mechanics, Executive Editor of Data-Centric Engineering, and Subject Editor for Applied Mathematical Modelling.

Affiliations and expertise

Research Unit in Eng Science, University of Luxembourg, Luxembourg

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Error Control, Adaptive Discretizations, and Applications, Part 1

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Franz Chouly

Stéphane P.A. Bordas