Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems

Carstensen, Carsten; Gallistl, Dietmar; Schedensack, Mira

doi:10.1090/S0025-5718-2014-02894-9

Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems
HTML articles powered by AMS MathViewer

by Carsten Carstensen, Dietmar Gallistl and Mira Schedensack PDF

Math. Comp. 84 (2015), 1061-1087 Request permission

Abstract:

The nonconforming approximation of eigenvalues is of high practical interest because it allows for guaranteed upper and lower eigenvalue bounds and for a convenient computation via a consistent diagonal mass matrix in 2D. The first main result is a comparison which states equivalence of the error of the nonconforming eigenvalue approximation with its best-approximation error and its error in a conforming computation on the same mesh. The second main result is optimality of an adaptive algorithm for the effective eigenvalue computation for the Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of a nonlinear approximation class. The analysis includes an inexact algebraic eigenvalue computation on each level of the adaptive algorithm which requires an iterative algorithm and a controlled termination criterion. The analysis is carried out for the first eigenvalue in a Laplace eigenvalue model problem in 2D.

References

I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
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 and Shipeng Mao, Convergence and quasi-optimal complexity of a simple adaptive finite element method, M2AN Math. Model. Numer. Anal. 43 (2009), no. 6, 1203–1219. MR 2588438, DOI 10.1051/m2an/2009036
Roland Becker and Shipeng Mao, Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations, SIAM J. Numer. Anal. 49 (2011), no. 3, 970–991. MR 2802555, DOI 10.1137/100802967
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
Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235, DOI 10.1017/CBO9780511618635
Dietrich Braess, An a posteriori error estimate and a comparison theorem for the nonconforming $P_1$ element, Calcolo 46 (2009), no. 2, 149–155. MR 2520373, DOI 10.1007/s10092-009-0003-z
S. C. Brenner and C. Carstensen, Encyclopedia of Computational Mechanics, ch. 4, Finite Element Methods, John Wiley and Sons, 2004.
Susanne C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods, Math. Comp. 65 (1996), no. 215, 897–921. MR 1348039, DOI 10.1090/S0025-5718-96-00746-6
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
C. Carstensen, M. Eigel, R. H. W. Hoppe, and C. Löbhard, A review of unified a posteriori finite element error control, Numer. Math. Theory Methods Appl. 5 (2012), no. 4, 509–558. MR 2968740, DOI 10.4208/nmtma.2011.m1032
Carsten Carstensen, Joscha Gedicke, and Donsub Rim, Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods, J. Comput. Math. 30 (2012), no. 4, 337–353. MR 2965987, DOI 10.4208/jcm.1108-m3677
C. Carstensen, D. Peterseim, and M. Schedensack, Comparison results of finite element methods for the Poisson model problem, SIAM J. Numer. Anal. 50 (2012), no. 6, 2803–2823. MR 3022243, DOI 10.1137/110845707
Carsten Carstensen and Hella Rabus, The adaptive nonconforming FEM for the pure displacement problem in linear elasticity is optimal and robust, SIAM J. Numer. Anal. 50 (2012), no. 3, 1264–1283. MR 2970742, DOI 10.1137/110824139
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 and Joscha Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comp. 83 (2014), no. 290, 2605–2629. MR 3246802, DOI 10.1090/S0025-5718-2014-02833-0
Carsten Carstensen and Ronald H. W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method, Numer. Math. 103 (2006), no. 2, 251–266. MR 2222810, DOI 10.1007/s00211-005-0658-6
Carsten Carstensen, Daniel Peterseim, and Hella Rabus, Optimal adaptive nonconforming FEM for the Stokes problem, Numer. Math. 123 (2013), no. 2, 291–308. MR 3010181, DOI 10.1007/s00211-012-0490-8
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
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
E. A. Dari, R. G. Durán, and C. Padra, A posteriori error estimates for non-conforming approximation of eigenvalue problems, Appl. Numer. Math. 62 (2012), no. 5, 580–591. MR 2899264, DOI 10.1016/j.apnum.2012.01.005
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
Thirupathi Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169–2189. MR 2684360, DOI 10.1090/S0025-5718-10-02360-4
Jun Hu and Jinchao Xu, Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem, J. Sci. Comput. 55 (2013), no. 1, 125–148. MR 3030706, DOI 10.1007/s10915-012-9625-4
ShiPeng Mao and ZhongCi Shi, On the error bounds of nonconforming finite elements, Sci. China Math. 53 (2010), no. 11, 2917–2926. MR 2736924, DOI 10.1007/s11425-010-3120-x
Shipeng Mao, Xuying Zhao, and Zhongci Shi, Convergence of a standard adaptive nonconforming finite element method with optimal complexity, Appl. Numer. Math. 60 (2010), no. 7, 673–688. MR 2646469, DOI 10.1016/j.apnum.2010.03.010
H. Rabus, A natural adaptive nonconforming FEM of quasi-optimal complexity, Comput. Methods Appl. Math. 10 (2010), no. 3, 315–325. MR 2770297, DOI 10.2478/cmam-2010-0018
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