Abstract
In this work, a space-time scheme for goal-oriented a posteriori error estimation is proposed. The error estimator is evaluated using a partition-of-unity dual-weighted residual method. As application, a low mach number combustion equation is considered. In some numerical tests, different interpolation variants are investigated, while observing convergence orders and effectivity indices between true errors (obtained on a sufficiently refined mesh) and the error estimator.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arndt, D., Bangerth, W., Blais, B., Clevenger, T.C., Fehling, M., Grayver, A.V., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Munch, P., Pelteret, J.P., Rastak, R., Tomas, I., Turcksin, B., Wang, Z., Wells, D.: The deal.II library, Version 9.2. J. Numer. Math. 28(3), 131–146 (2020)
Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001)
Besier, M.: Adaptive finite element methods for computing nonstationary incompressible flows. Ph.D. thesis, University of Heidelberg (2009)
Besier, M., Rannacher, R.: Goal-oriented space-time adaptivity in the finite element galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Methods Fluids 70, 1139–1166 (2012)
Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic obstacle problems. Electron. Trans. Numer. Anal. 32, 162–172 (2008)
Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic signorini problems. Comput. Mech. 44(4), 481–491 (2009)
Braack, M., Ern, A.: A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003)
Endtmayer, B., Langer, U., Wick, T.: Reliability and efficiency of DWR-type a posteriori error estimates with smart sensitivity weight recovering. Comput. Methods Appl. Math. 21(2), 351 (2021)
Failer, L.: Optimal control of time-dependent nonlinear fluid-structure interaction. Ph.D. thesis, Technical University Munich (2017)
Failer, L., Wick, T.: Adaptive time-step control for nonlinear fluid-structure interaction. J. Comput. Phys. 366, 448–477 (2018)
Köcher, U., Bruchhäuser, M.P., Bause, M.: Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space–time FEM codes. SoftwareX 10, 100239 (2019)
Rademacher, A.: Adaptive finite element methods for nonlinear hyperbolic problems of second order. Ph.D. thesis, Technische Universität Dortmund (2009)
Richter, T., Wick, T.: Variational localizations of the dual weighted residual estimator. J. Comput. Appl. Math. 279(0), 192–208 (2015)
Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2008)
Thiele, J., Wick, T.: Space-time PU-DWR error control and adaptivity for the heat equation. Proc. Appl. Math. Mech. 21(1), e202100174 (2021)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Thiele, J.P., Wick, T. (2023). Space-Time Error Control Using a Partition-of-Unity Dual-Weighted Residual Method Applied to Low Mach Number Combustion. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_34
Download citation
DOI: https://doi.org/10.1007/978-3-031-20432-6_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20431-9
Online ISBN: 978-3-031-20432-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)