Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form

Boffi, Daniele; Gallistl, Dietmar; Gardini, Francesca; Gastaldi, Lucia

doi:10.1090/mcom/3212

Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form
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by Daniele Boffi, Dietmar Gallistl, Francesca Gardini and Lucia Gastaldi PDF

Math. Comp. 86 (2017), 2213-2237 Request permission

Abstract:

It is shown that the $h$-adaptive mixed finite element method for the discretization of eigenvalue clusters of the Laplace operator produces optimal convergence rates in terms of nonlinear approximation classes. The results are valid for the typical mixed spaces of Raviart–Thomas or Brezzi–Douglas–Marini type with arbitrary fixed polynomial degree in two and three space dimensions.

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, 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
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
Roland Becker, Shipeng Mao, and Zhongci Shi, A convergent nonconforming adaptive finite element method with quasi-optimal complexity, SIAM J. Numer. Anal. 47 (2010), no. 6, 4639–4659. MR 2595052, DOI 10.1137/070701479
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
Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. MR 2652780, DOI 10.1017/S0962492910000012
Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
Daniele Boffi, Franco Brezzi, and Lucia Gastaldi, On the convergence of eigenvalues for mixed formulations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 131–154 (1998). Dedicated to Ennio De Giorgi. MR 1655512
Andrea Bonito and Alan Demlow, Convergence and optimality of higher-order adaptive finite element methods for eigenvalue clusters, SIAM J. Numer. Anal. 54 (2016), no. 4, 2379–2388. MR 3532806, DOI 10.1137/15M1036877
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
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
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
Carsten Carstensen, Dietmar Gallistl, and Mira Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comp. 84 (2015), no. 293, 1061–1087. MR 3315500, DOI 10.1090/S0025-5718-2014-02894-9
Carsten Carstensen and Joscha Gedicke, An oscillation-free adaptive FEM for symmetric eigenvalue problems, Numer. Math. 118 (2011), no. 3, 401–427. MR 2810801, DOI 10.1007/s00211-011-0367-2
Carsten Carstensen and Joscha Gedicke, An adaptive finite element eigenvalue solver of asymptotic quasi-optimal computational complexity, SIAM J. Numer. Anal. 50 (2012), no. 3, 1029–1057. MR 2970733, DOI 10.1137/090769430
Carsten Carstensen, Daniel Peterseim, and Andreas Schröder, The norm of a discretized gradient in $H(\mathrm {div})^*$ for a posteriori finite element error analysis, Numer. Math. 132 (2016), no. 3, 519–539. MR 3457439, DOI 10.1007/s00211-015-0728-3
Carsten Carstensen and Hella Rabus, An optimal adaptive mixed finite element method, Math. Comp. 80 (2011), no. 274, 649–667. MR 2772091, DOI 10.1090/S0025-5718-2010-02397-X
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
Xiaoying Dai, Lianhua He, and Aihui Zhou, Convergence and quasi-optimal complexity of adaptive finite element computations for multiple eigenvalues, IMA J. Numer. Anal. 35 (2015), no. 4, 1934–1977. MR 3407250, DOI 10.1093/imanum/dru059
Xiaoying Dai, Jinchao Xu, and Aihui Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math. 110 (2008), no. 3, 313–355. MR 2430983, DOI 10.1007/s00211-008-0169-3
Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. II. Error estimates for the Galerkin method, RAIRO Anal. Numér. 12 (1978), no. 2, 113–119, iii (English, with French summary). MR 483401, DOI 10.1051/m2an/1978120201131
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
J. Douglas Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103 (English, with Portuguese summary). MR 667620
Ricardo G. Durán, Lucia Gastaldi, and Claudio Padra, A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Methods Appl. Sci. 9 (1999), no. 8, 1165–1178. MR 1722056, DOI 10.1142/S021820259900052X
D. Gallistl, Adaptive finite element computation of eigenvalues, Doctoral dissertation, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, 2014.
Dietmar Gallistl, Adaptive nonconforming finite element approximation of eigenvalue clusters, Comput. Methods Appl. Math. 14 (2014), no. 4, 509–535. MR 3259027, DOI 10.1515/cmam-2014-0020
Dietmar Gallistl, An optimal adaptive FEM for eigenvalue clusters, Numer. Math. 130 (2015), no. 3, 467–496. MR 3347459, DOI 10.1007/s00211-014-0671-8
Eduardo M. Garau, Pedro Morin, and Carlos Zuppa, Convergence of adaptive finite element methods for eigenvalue problems, Math. Models Methods Appl. Sci. 19 (2009), no. 5, 721–747. MR 2531037, DOI 10.1142/S0218202509003590
Francesca Gardini, Mixed approximation of eigenvalue problems: a superconvergence result, M2AN Math. Model. Numer. Anal. 43 (2009), no. 5, 853–865. MR 2559736, DOI 10.1051/m2an/2009005
Gabriel N. Gatica and Matthias Maischak, A posteriori error estimates for the mixed finite element method with Lagrange multipliers, Numer. Methods Partial Differential Equations 21 (2005), no. 3, 421–450. MR 2128589, DOI 10.1002/num.20050
Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas, A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes-Darcy coupled problem, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 21-22, 1877–1891. MR 2787543, DOI 10.1016/j.cma.2011.02.009
S. Giani and I. G. Graham, A convergent adaptive method for elliptic eigenvalue problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1067–1091. MR 2485445, DOI 10.1137/070697264
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
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
Mats G. Larson and Axel Målqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math. 108 (2008), no. 3, 487–500. MR 2365826, DOI 10.1007/s00211-007-0121-y
Carlo Lovadina and Rolf Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75 (2006), no. 256, 1659–1674. MR 2240629, DOI 10.1090/S0025-5718-06-01872-2
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
Ricardo H. Nochetto and Andreas Veeser, Primer of adaptive finite element methods, Multiscale and adaptivity: modeling, numerics and applications, Lecture Notes in Math., vol. 2040, Springer, Heidelberg, 2012, pp. 125–225. MR 3076038, DOI 10.1007/978-3-642-24079-9
J. E. Pasciak and J. Zhao, Overlapping Schwarz methods in $H$(curl) on polyhedral domains, J. Numer. Math. 10 (2002), no. 3, 221–234. MR 1935967, DOI 10.1515/JNMA.2002.221
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
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Barbara I. Wohlmuth and Ronald H. W. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Math. Comp. 68 (1999), no. 228, 1347–1378. MR 1651760, DOI 10.1090/S0025-5718-99-01125-4