Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors

Li, Yuwen

doi:10.1090/mcom/3590

Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors
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Math. Comp. 90 (2021), 565-593 Request permission

Abstract:

For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic forms is trivial. In particular, we obtain new quasi-optimal adaptive mixed methods for the Hodge Laplace, Poisson, and Stokes equations. Comparing to existing adaptive mixed methods, the new methods control errors in both variables.

References

A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
Douglas N. Arnold, Jim Douglas Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. MR 761879, DOI 10.1007/BF01379659
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math. 44 (1984), no. 1, 75–102. MR 745088, DOI 10.1007/BF01389757
Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
Roland Becker and Shipeng Mao, An optimally convergent adaptive mixed finite element method, Numer. Math. 111 (2008), no. 1, 35–54. MR 2448202, DOI 10.1007/s00211-008-0180-8
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
Peter Binev, Wolfgang Dahmen, Ronald DeVore, and Pencho Petrushev, Approximation classes for adaptive methods, Serdica Math. J. 28 (2002), no. 4, 391–416. Dedicated to the memory of Vassil Popov on the occasion of his 60th birthday. MR 1965238
Peter Binev and Ronald DeVore, Fast computation in adaptive tree approximation, Numer. Math. 97 (2004), no. 2, 193–217. MR 2050076, DOI 10.1007/s00211-003-0493-6
D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431–2444. MR 1427472, DOI 10.1137/S0036142994264079
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Zhiqiang Cai, Charles Tong, Panayot S. Vassilevski, and Chunbo Wang, Mixed finite element methods for incompressible flow: stationary Stokes equations, Numer. Methods Partial Differential Equations 26 (2010), no. 4, 957–978. MR 2642330, DOI 10.1002/num.20467
Zhiqiang Cai, Chunbo Wang, and Shun Zhang, Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations, SIAM J. Numer. Anal. 48 (2010), no. 1, 79–94. MR 2608359, DOI 10.1137/080718413
Zhiqiang Cai and Yanqiu Wang, A multigrid method for the pseudostress formulation of Stokes problems, SIAM J. Sci. Comput. 29 (2007), no. 5, 2078–2095. MR 2350022, DOI 10.1137/060661429
C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. MR 3170325, DOI 10.1016/j.camwa.2013.12.003
C. Carstensen and H. Rabus, Axioms of adaptivity with separate marking for data resolution, SIAM J. Numer. Anal. 55 (2017), no. 6, 2644–2665. MR 3719030, DOI 10.1137/16M1068050
Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
Carsten Carstensen, Asha K. Dond, and Hella Rabus, Quasi-optimality of adaptive mixed FEMs for non-selfadjoint indefinite second-order linear elliptic problems, Comput. Methods Appl. Math. 19 (2019), no. 2, 233–250. MR 3935887, DOI 10.1515/cmam-2019-0034
Carsten Carstensen, Dietmar Gallistl, and Mira Schedensack, Quasi-optimal adaptive pseudostress approximation of the Stokes equations, SIAM J. Numer. Anal. 51 (2013), no. 3, 1715–1734. MR 3066804, DOI 10.1137/110852346
Carsten Carstensen and R. H. W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method, Math. Comp. 75 (2006), no. 255, 1033–1042. MR 2219017, DOI 10.1090/S0025-5718-06-01829-1
Carsten Carstensen, Dongho Kim, and Eun-Jae Park, A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem, SIAM J. Numer. Anal. 49 (2011), no. 6, 2501–2523. MR 2873244, DOI 10.1137/100816237
Carsten Carstensen and Eun-Jae Park, Convergence and optimality of adaptive least squares finite element methods, SIAM J. Numer. Anal. 53 (2015), no. 1, 43–62. MR 3296614, DOI 10.1137/130949634
J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046, DOI 10.1137/07069047X
Long Chen, Michael Holst, and Jinchao Xu, Convergence and optimality of adaptive mixed finite element methods, Math. Comp. 78 (2009), no. 265, 35–53. MR 2448696, DOI 10.1090/S0025-5718-08-02155-8
Long Chen and Yongke Wu, Convergence of adaptive mixed finite element methods for the Hodge Laplacian equation: without harmonic forms, SIAM J. Numer. Anal. 55 (2017), no. 6, 2905–2929. MR 3725282, DOI 10.1137/16M1097912
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, DOI 10.1090/S0025-5718-07-02081-9
Albert Cohen, Ronald DeVore, and Ricardo H. Nochetto, Convergence rates of AFEM with $H^{-1}$ data, Found. Comput. Math. 12 (2012), no. 5, 671–718. MR 2970853, DOI 10.1007/s10208-012-9120-1
Alan Demlow, Convergence and quasi-optimality of adaptive finite element methods for harmonic forms, Numer. Math. 136 (2017), no. 4, 941–971. MR 3671593, DOI 10.1007/s00211-016-0862-6
Alan Demlow and Anil N. Hirani, A posteriori error estimates for finite element exterior calculus: the de Rham complex, Found. Comput. Math. 14 (2014), no. 6, 1337–1371. MR 3273681, DOI 10.1007/s10208-014-9203-2
Lars Diening, Christian Kreuzer, and Rob Stevenson, Instance optimality of the adaptive maximum strategy, Found. Comput. Math. 16 (2016), no. 1, 33–68. MR 3451423, DOI 10.1007/s10208-014-9236-6
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Richard S. Falk and Ragnar Winther, Local bounded cochain projections, Math. Comp. 83 (2014), no. 290, 2631–2656. MR 3246803, DOI 10.1090/S0025-5718-2014-02827-5
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. MR 3176325, DOI 10.1137/120897225
Michael Feischl, Optimality of a standard adaptive finite element method for the Stokes problem, SIAM J. Numer. Anal. 57 (2019), no. 3, 1124–1157. MR 3953464, DOI 10.1137/17M1153170
Matthew P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc. 78 (1955), 426–444. MR 68888, DOI 10.1090/S0002-9947-1955-0068888-1
Fernando D. Gaspoz and Pedro Morin, Approximation classes for adaptive higher order finite element approximation, Math. Comp. 83 (2014), no. 289, 2127–2160. MR 3223327, DOI 10.1090/S0025-5718-2013-02777-9
Michael Holst, Yuwen Li, Adam Mihalik, and Ryan Szypowski, Convergence and optimality of adaptive mixed methods for Poisson’s equation in the FEEC framework, J. Comput. Math. 38 (2020), no. 5, 748–767. MR 4091623, DOI 10.4208/jcm.1905-m2018-0265
Jun Hu and Guozhu Yu, A unified analysis of quasi-optimal convergence for adaptive mixed finite element methods, SIAM J. Numer. Anal. 56 (2018), no. 1, 296–316. MR 3749387, DOI 10.1137/16M105513X
JianGuo Huang and YiFeng Xu, Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation, Sci. China Math. 55 (2012), no. 5, 1083–1098. MR 2912497, DOI 10.1007/s11425-012-4384-0
Yuwen Li, Some convergence and optimality results of adaptive mixed methods in finite element exterior calculus, SIAM J. Numer. Anal. 57 (2019), no. 4, 2019–2042. MR 3995302, DOI 10.1137/18M1229080
Joseph M. Maubach, Local bisection refinement for $n$-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), no. 1, 210–227. MR 1311687, DOI 10.1137/0916014
Dorina Mitrea, Marius Mitrea, and Sylvie Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal. 7 (2008), no. 6, 1295–1333. MR 2425010, DOI 10.3934/cpaa.2008.7.1295
Dorina Mitrea, Marius Mitrea, and Mei-Chi Shaw, Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions, Indiana Univ. Math. J. 57 (2008), no. 5, 2061–2095. MR 2463962, DOI 10.1512/iumj.2008.57.3338
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Pedro Morin, Kunibert G. Siebert, and Andreas Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 707–737. MR 2413035, DOI 10.1142/S0218202508002838
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Ricardo H. Nochetto, Kunibert G. Siebert, and Andreas Veeser, Theory of adaptive finite element methods: an introduction, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 409–542. MR 2648380, DOI 10.1007/978-3-642-03413-8_{1}2
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, DOI 10.1090/S0025-5718-07-02030-3
Günter Schwarz, Hodge decomposition—a method for solving boundary value problems, Lecture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR 1367287, DOI 10.1007/BFb0095978
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X
C. T. Traxler, An algorithm for adaptive mesh refinement in $n$ dimensions, Computing 59 (1997), no. 2, 115–137. MR 1475530, DOI 10.1007/BF02684475
Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. MR 3059294, DOI 10.1093/acprof:oso/9780199679423.001.0001
Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum, and Jinchao Xu, Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations, Math. Comp. 81 (2012), no. 278, 623–642. MR 2869030, DOI 10.1090/S0025-5718-2011-02544-5