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Instance-Optimal Goal-Oriented Adaptivity

  • Michael Innerberger EMAIL logo and Dirk Praetorius


We consider an adaptive finite element method with arbitrary but fixed polynomial degree p 1 , where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 2016, 1, 33–68], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

Funding source: Austrian Science Fund

Award Identifier / Grant number: W1245

Award Identifier / Grant number: SFB F65

Award Identifier / Grant number: P27005

Funding statement: The authors acknowledge support through the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), the special research program Taming complexity in PDE systems (grant SFB F65), and the stand-alone project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).


[1] R. Becker, E. Estecahandy and D. Trujillo, Weighted marking for goal-oriented adaptive finite element methods, SIAM J. Numer. Anal. 49 (2011), no. 6, 2451–2469. 10.1137/100794298Search in Google Scholar

[2] A. Bespalov, A. Haberl and D. Praetorius, Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems, Comput. Methods Appl. Mech. Engrg. 317 (2017), 318–340. 10.1016/j.cma.2016.12.014Search in Google Scholar

[3] P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. 10.21236/ADA640658Search in Google Scholar

[4] P. Binev, W. Dahmen, R. DeVore and P. Petrushev, Approximation classes for adaptive methods, Serdica Math. J. 28 (2002), no. 4, 391–416. Search in Google Scholar

[5] C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. 10.1016/j.camwa.2013.12.003Search in Google Scholar PubMed PubMed Central

[6] J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. 10.1137/07069047XSearch in Google Scholar

[7] L. Diening, C. Kreuzer and R. Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 (2016), no. 1, 33–68. 10.1007/s10208-014-9236-6Search in Google Scholar

[8] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. 10.1137/0733054Search in Google Scholar

[9] C. Erath, G. Gantner and D. Praetorius, Optimal convergence behavior of adaptive FEM driven by simple ( h - h / 2 ) -type error estimators, Comput. Math. Appl. 79 (2020), no. 3, 623–642. 10.1016/j.camwa.2019.07.014Search in Google Scholar

[10] M. Feischl, T. Führer and D. Praetorius, Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems, SIAM J. Numer. Anal. 52 (2014), no. 2, 601–625. 10.1137/120897225Search in Google Scholar

[11] M. Feischl, G. Gantner, A. Haberl, D. Praetorius and T. Führer, Adaptive boundary element methods for optimal convergence of point errors, Numer. Math. 132 (2016), no. 3, 541–567. 10.1007/s00211-015-0727-4Search in Google Scholar

[12] M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal. 54 (2016), no. 3, 1423–1448. 10.1137/15M1021982Search in Google Scholar

[13] S. Ferraz-Leite, C. Ortner and D. Praetorius, Convergence of simple adaptive Galerkin schemes based on h - h / 2 error estimators, Numer. Math. 116 (2010), no. 2, 291–316. 10.1007/s00211-010-0292-9Search in Google Scholar

[14] T. Führer, S. A. Funken and D. Praetorius, Adaptive isoparametric P2-FEM: Analysis and efficient Matlab implementation, in preparation. Search in Google Scholar

[15] S. Funken, D. Praetorius and P. Wissgott, Efficient implementation of adaptive P1-FEM in Matlab, Comput. Methods Appl. Math. 11 (2011), no. 4, 460–490. 10.2478/cmam-2011-0026Search in Google Scholar

[16] T. Gantumur, Convergence rates of adaptive methods, Besov spaces, and multilevel approximation, Found. Comput. Math. 17 (2017), no. 4, 917–956. 10.1007/s10208-016-9308-xSearch in Google Scholar

[17] F. D. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element approximation, Math. Comp. 83 (2014), no. 289, 2127–2160. 10.1090/S0025-5718-2013-02777-9Search in Google Scholar

[18] M. Holst and S. Pollock, Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems, Numer. Methods Partial Differential Equations 32 (2016), no. 2, 479–509. 10.1002/num.22002Search in Google Scholar

[19] M. Holst, S. Pollock and Y. Zhu, Convergence of goal-oriented adaptive finite element methods for semilinear problems, Comput. Vis. Sci. 17 (2015), no. 1, 43–63. 10.1007/s00791-015-0243-1Search in Google Scholar

[20] M. Karkulik, D. Pavlicek and D. Praetorius, On 2D newest vertex bisection: Optimality of mesh-closure and H 1 -stability of L 2 -projection, Constr. Approx. 38 (2013), no. 2, 213–234. 10.1007/s00365-013-9192-4Search in Google Scholar

[21] C. Kreuzer and M. Schedensack, Instance optimal Crouzeix–Raviart adaptive finite element methods for the Poisson and Stokes problems, IMA J. Numer. Anal. 36 (2016), no. 2, 593–617. 10.1093/imanum/drv019Search in Google Scholar

[22] M. S. Mommer and R. Stevenson, A goal-oriented adaptive finite element method with convergence rates, SIAM J. Numer. Anal. 47 (2009), no. 2, 861–886. 10.1137/060675666Search in Google Scholar

[23] P. Morin, R. H. Nochetto and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. 10.1137/S0036142999360044Search in Google Scholar

[24] P. Morin, K. G. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 707–737. 10.1142/S0218202508002838Search in Google Scholar

[25] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. 10.1090/S0025-5718-1990-1011446-7Search in Google Scholar

[26] K. G. Siebert, A convergence proof for adaptive finite elements without lower bound, IMA J. Numer. Anal. 31 (2011), no. 3, 947–970. 10.1093/imanum/drq001Search in Google Scholar

[27] R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. 10.1007/s10208-005-0183-0Search in Google Scholar

[28] R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. 10.1090/S0025-5718-07-01959-XSearch in Google Scholar

Received: 2019-07-31
Revised: 2019-12-17
Accepted: 2020-02-02
Published Online: 2020-02-18
Published in Print: 2021-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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