Details
| Original language | English |
|---|---|
| Title of host publication | Error Control, Adaptive Discretizations, and Applications, Part 2 |
| Publisher | Academic Press Inc. |
| Pages | 19-108 |
| Number of pages | 90 |
| ISBN (print) | 9780443294501 |
| Publication status | Published - 2024 |
Publication series
| Name | Advances in Applied Mechanics |
|---|---|
| Volume | 59 |
| ISSN (Print) | 0065-2156 |
Abstract
This chapter reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic p-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.
Keywords
- Adaptive finite element methods, Adjoint problems, Dual-weighted residual method, Goal-oriented error control, Multi-goal error control, Partial differential equations
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Engineering(all)
- Mechanical Engineering
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Error Control, Adaptive Discretizations, and Applications, Part 2. Academic Press Inc., 2024. p. 19-108 (Advances in Applied Mechanics; Vol. 59).
Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review
}
TY - CHAP
T1 - A posteriori single- and multi-goal error control and adaptivity for partial differential equations
AU - Endtmayer, Bernhard
AU - Langer, Ulrich
AU - Richter, Thomas
AU - Schafelner, Andreas
AU - Wick, Thomas
N1 - Publisher Copyright: © 2024
PY - 2024
Y1 - 2024
N2 - This chapter reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic p-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.
AB - This chapter reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic p-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.
KW - Adaptive finite element methods
KW - Adjoint problems
KW - Dual-weighted residual method
KW - Goal-oriented error control
KW - Multi-goal error control
KW - Partial differential equations
UR - http://www.scopus.com/inward/record.url?scp=85191584742&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2404.01738
DO - 10.48550/arXiv.2404.01738
M3 - Contribution to book/anthology
AN - SCOPUS:85191584742
SN - 9780443294501
T3 - Advances in Applied Mechanics
SP - 19
EP - 108
BT - Error Control, Adaptive Discretizations, and Applications, Part 2
PB - Academic Press Inc.
ER -