Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 12, 2017

The Dual-Weighted Residual Estimator Realized on Polygonal Meshes

  • Steffen Weißer ORCID logo EMAIL logo and Thomas Wick

Abstract

In this work, we realize goal-oriented error estimation using the dual-weighted residual method on general polygonal meshes. Such meshes are of current interest in various applications thanks to their great flexibility. Specifically the discrete problems are treated on BEM-based FEM. Our dual-weighted residual estimator is derived for two localization procedures. Firstly, a classical (strong) localization. Secondly, a weak form is adopted in which localization is achieved with the help of a partition-of-unity. The dual (i.e., adjoint) solution is obtained via a local higher-order approximation using a single element. Our algorithmic developments are substantiated with the help of several numerical tests.

MSC 2010: 65N30; 65N38; 65N50

References

[1] M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg. 142 (1997), no. 1–2, 1–88. 10.1002/9781118032824Search in Google Scholar

[2] M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (New York), John Wiley & Sons, New York, 2000. 10.1002/9781118032824Search in Google Scholar

[3] P. F. Antonietti, L. B. da Veiga, C. Lovadina and M. Verani, Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 654–675. 10.1137/120873157Search in Google Scholar

[4] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lect. Math. ETH Zürich, Birkhäuser, Basel, 2003. 10.1007/978-3-0348-7605-6Search in Google Scholar

[5] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts, SIAM J. Optim. Control 39 (2000), 113–132. 10.1137/S0363012999351097Search in Google Scholar

[6] R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, ENUMATH’97—Proceedings of the 2nd European Conference on Numerical Mathematics and Advanced Applications (Heidelberg 1997), World Scientific, Singapore (1995), 18–22. Search in Google Scholar

[7] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001), 1–102. 10.1017/S0962492901000010Search in Google Scholar

[8] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul. 1 (2003), no. 2, 221–238. 10.1137/S1540345902410482Search in Google Scholar

[9] A. Cangiani, E. H. Georgoulis, T. Prayer and O. J. Sutton, A posteriori error estimates for the Virtual Element Method, Numer. Math. (2017), 10.1007/s00211-017-0891-9. 10.1007/s00211-017-0891-9Search in Google Scholar PubMed PubMed Central

[10] C. Carstensen, Estimation of higher sobolev norm from lower order approximation, SIAM J. Numer. Anal. 42 (2004), 2136–2147. 10.1137/S0036142902413615Search in Google Scholar

[11] C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571–1587. 10.1137/S003614299732334XSearch in Google Scholar

[12] L. Chen, J. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput. 59 (2014), no. 2, 496–511. 10.1007/s10915-013-9771-3Search in Google Scholar

[13] D. Copeland, U. Langer and D. Pusch, From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes, Domain Decomposition Methods in Science and Engineering XVIII, Lect. Notes Comput. Sci. Eng. 70, Springer, Berlin (2009), 315–322. 10.1007/978-3-642-02677-5_35Search in Google Scholar

[14] L. B. da Veiga and G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal. 49 (2015), no. 2, 577–599. 10.1051/m2an/2014047Search in Google Scholar

[15] B. Endtmayer, Adaptive mesh refinement for multiple goal functionals, Master’s thesis, Johannes Kepler University Linz, Institute of Computational Mathematics, 2017. Search in Google Scholar

[16] B. Endtmayer and T. Wick, A partition-of-unity dual-weighted residual approach for multi-objective goal functional error estimation applied to elliptic problems, Comput. Methods Appl. Math. 17 (2017), no. 4, 575–599.10.1515/cmam-2017-0001Search in Google Scholar

[17] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica 1995, Cambridge University Press, Cambridge (1995), 105–158. 10.1017/S0962492900002531Search in Google Scholar

[18] A. L. Gain, C. Talischi and G. H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg. 282 (2014), 132–160. 10.1016/j.cma.2014.05.005Search in Google Scholar

[19] M. B. Giles and E. Süli, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numerica 11 (2002), 145–236. 10.1017/CBO9780511550140.003Search in Google Scholar

[20] C. Hofreither, U. Langer and S. Weißer, Convection adapted BEM-based FEM, ZAMM Z. Angew. Math. Mech. 96 (2016), no. 12, 1467–1481. 10.1002/zamm.201500042Search in Google Scholar

[21] O. A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. 10.1137/S0036142902405217Search in Google Scholar

[22] G. Kuru, C. V. Verhoosel, K. G. van der Zee and E. H. van Brummelen, Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Engrg. 270 (2014), 270–292. 10.1016/j.cma.2013.11.026Search in Google Scholar

[23] D. Kuzmin and S. Korotov, Goal-oriented a posteriori error estimates for transport problems, Math. Comput. Simulation 80 (2010), no. 8, 1674–1683. 10.1016/j.matcom.2009.03.008Search in Google Scholar

[24] R. Lazarov, S. Repin and S. Tomar, Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems, Numer. Methods Partial Differential Equations 25 (2009), no. 4, 952–971. 10.1002/num.20386Search in Google Scholar

[25] G. Manzini, A. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1665–1699. 10.1142/S0218202514400065Search in Google Scholar

[26] W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Search in Google Scholar

[27] R. H. Nochetto, A. Veeser and M. Verani, A safeguarded dual weighted residual method, IMA J. Numer. Anal. 29 (2009), no. 1, 126–140. 10.1093/imanum/drm026Search in Google Scholar

[28] J. T. Oden and S. Prudhomme, On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg. 176 (1999), 313–331. 10.1016/S0045-7825(98)00343-0Search in Google Scholar

[29] J. Peraire and A. T. Patera, Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement, Advances in Adaptive Computational Methods in Mechanics, Elsevier, Amsterdam (1998), 199–215. 10.1016/S0922-5382(98)80011-1Search in Google Scholar

[30] D. A. D. Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. 10.1016/j.cma.2014.09.009Search in Google Scholar

[31] R. Rannacher and F.-T. Suttmeier, A feed-back approach to error control in finite element methods: Application to linear elasticity, Comput. Mech. 19 (1997), no. 5, 434–446. 10.1007/s004660050191Search in Google Scholar

[32] R. Rannacher and F.-T. Suttmeier, A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity, Comput. Methods Appl. Mech. Engrg. 176 (1999), no. 1–4, 333–361. 10.1016/S0045-7825(98)00344-2Search in Google Scholar

[33] T. Richter, Goal-oriented error estimation for fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg. 223–224 (2012), 38–42. 10.1016/j.cma.2012.02.014Search in Google Scholar

[34] T. Richter and T. Wick, Variational localizations of the dual weighted residual estimator, J. Comput. Appl. Math. 279 (2015), 192–208. 10.1016/j.cam.2014.11.008Search in Google Scholar

[35] S. Rjasanow and S. Weißer, Higher order BEM-based FEM on polygonal meshes, SIAM J. Numer. Anal. 50 (2012), no. 5, 2357–2378. 10.1137/110849481Search in Google Scholar

[36] S. Rjasanow and S. Weißer, FEM with Trefftz trial functions on polyhedral elements, J. Comput. Appl. Math. 263 (2014), 202–217. 10.1016/j.cam.2013.12.023Search in Google Scholar

[37] A. Schroeder and A. Rademacher, Goal-oriented error control in adaptive mixed FEM for Signorini’s problem, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 1–4, 345–355. 10.1016/j.cma.2010.08.015Search in Google Scholar

[38] R. Scott, Optimal L estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. 10.1090/S0025-5718-1976-0436617-2Search in Google Scholar

[39] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements, Springer, New York, 2007. 10.1007/978-0-387-68805-3Search in Google Scholar

[40] F.-T. Suttmeier, Numerical solution of Variational Inequalities by Adaptive Finite Elements, Vieweg+Teubner, Wiesbaden, 2008. Search in Google Scholar

[41] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley/Teubner, New York/Stuttgart, 1996. Search in Google Scholar

[42] S. Weißer, Residual error estimate for BEM-based FEM on polygonal meshes, Numer. Math. 118 (2011), no. 4, 765–788. 10.1007/s00211-011-0371-6Search in Google Scholar

[43] S. Weißer, Finite Element Methods with local Trefftz trial functions, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, 2012. Search in Google Scholar

[44] S. Weißer, Arbitrary order Trefftz-like basis functions on polygonal meshes and realization in BEM-based FEM, Comput. Math. Appl. 67 (2014), no. 7, 1390–1406. 10.1016/j.camwa.2014.01.019Search in Google Scholar

[45] S. Weißer, Residual based error estimate for higher order Trefftz-like trial functions on adaptively refined polygonal meshes, Numerical Mathematics and Advanced Applications—ENUMATH 2013, Lect. Notes Comput. Sci. Eng. 103, Springer, Cham (2015), 233–241. 10.1007/978-3-319-10705-9_23Search in Google Scholar

[46] S. Weißer, Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM, Comput. Math. Appl. 73 (2017), no. 2, 187–202. 10.1016/j.camwa.2016.11.013Search in Google Scholar

[47] T. Wick, Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity, Comput. Mech. 57 (2016), no. 6, 1017–1035. 10.1007/s00466-016-1275-1Search in Google Scholar

[48] K. Zee, E. Brummelen, I. Akkerman and R. Borst, Goal-oriented error estimation and adaptivity for fluid-structure interaction using exact linearized adjoints, Comput. Methods Appl. Mech. Engrg. 200 (2011), 2738–2757. 10.1016/j.cma.2010.12.010Search in Google Scholar

Received: 2017-04-25
Revised: 2017-09-01
Accepted: 2017-10-03
Published Online: 2017-11-12
Published in Print: 2018-10-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.4.2024 from https://www.degruyter.com/document/doi/10.1515/cmam-2017-0046/html
Scroll to top button