Inf-sup stability implies quasi-orthogonality

Feischl, Michael

doi:10.1090/mcom/3748

Inf-sup stability implies quasi-orthogonality
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Math. Comp. 91 (2022), 2059-2094 Request permission

Abstract:

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tools are new stability bounds for the $LU$-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
George A. Anastassiou, Poincaré and Sobolev type inequalities for vector-valued functions, Comput. Math. Appl. 56 (2008), no. 4, 1102–1113. MR 2437880, DOI 10.1016/j.camwa.2008.02.012
Kevin T. Andrews and Joseph D. Ward, $LU$-factorization of order bounded operators on Banach sequence spaces, J. Approx. Theory 48 (1986), no. 2, 169–180. MR 862232, DOI 10.1016/0021-9045(86)90001-8
M. Aurada, M. Feischl, M. Karkulik, and D. Praetorius, A posteriori error estimates for the Johnson-Nédélec FEM-BEM coupling, Eng. Anal. Bound. Elem. 36 (2012), no. 2, 255–266. MR 2847121, DOI 10.1016/j.enganabound.2011.07.017
Markus Aurada, Michael Feischl, Thomas Führer, Michael Karkulik, Jens Markus Melenk, and Dirk Praetorius, Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity, Comput. Mech. 51 (2013), no. 4, 399–419. MR 3038165, DOI 10.1007/s00466-012-0779-6
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J. M. Melenk, and D. Praetorius, Local inverse estimates for non-local boundary integral operators, Math. Comp. 86 (2017), no. 308, 2651–2686. MR 3667020, DOI 10.1090/mcom/3175
Markus Aurada, Michael Feischl, and Dirk Praetorius, Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 5, 1147–1173. MR 2916376, DOI 10.1051/m2an/2011075
I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1–40. MR 880421, DOI 10.1016/0045-7825(87)90114-9
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
Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
Rajendra Bhatia, Pinching, trimming, truncating, and averaging of matrices, Amer. Math. Monthly 107 (2000), no. 7, 602–608. MR 1786234, DOI 10.2307/2589115
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, Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal. 34 (1997), no. 2, 664–670. MR 1442933, DOI 10.1137/S0036142994270193
Andrea Bonito and Ricardo H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48 (2010), no. 2, 734–771. MR 2670003, DOI 10.1137/08072838X
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
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, 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
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
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
J. Manuel Cascón and Ricardo H. Nochetto, Quasioptimal cardinality of AFEM driven by nonresidual estimators, IMA J. Numer. Anal. 32 (2012), no. 1, 1–29. MR 2875241, DOI 10.1093/imanum/drr014
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
Martin Costabel, A symmetric method for the coupling of finite elements and boundary elements, The mathematics of finite elements and applications, VI (Uxbridge, 1987) Academic Press, London, 1988, pp. 281–288. MR 956899
Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075, DOI 10.1007/978-3-642-58090-1
E. B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. (2) 37 (1988), no. 1, 148–157. MR 921753, DOI 10.1112/jlms/s2-37.121.148
Mikael de la Salle (https://mathoverflow.net/users/10265/mikael-de-lasalle), Norm of triangular truncation operator on rank deficient matrices. https://mathoverflow.net/q/181917 (version: 2014-09-30).
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
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1009163
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Michael Feischl, Optimal adaptivity for non-symmetric fem/bem coupling, arXiv:1710.06082 (2017).
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
Michael Feischl, Thomas Führer, Michael Karkulik, Jens Markus Melenk, and Dirk Praetorius, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation, Calcolo 51 (2014), no. 4, 531–562. MR 3279576, DOI 10.1007/s10092-013-0100-x
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
M. Feischl, M. Karkulik, J. M. Melenk, and D. Praetorius, Quasi-optimal convergence rate for an adaptive boundary element method, SIAM J. Numer. Anal. 51 (2013), no. 2, 1327–1348. MR 3047442, DOI 10.1137/110842569
Stefan Funken, Dirk Praetorius, and Philipp Wissgott, Efficient implementation of adaptive P1-FEM in Matlab, Comput. Methods Appl. Math. 11 (2011), no. 4, 460–490. MR 2875100, DOI 10.2478/cmam-2011-0026
Gregor Gantner, Alexander Haberl, Dirk Praetorius, and Stefan Schimanko, Rate optimality of adaptive finite element methods with respect to overall computational costs, Math. Comp. 90 (2021), no. 331, 2011–2040. MR 4280291, DOI 10.1090/mcom/3654
Tsogtgerel Gantumur, On the convergence theory of adaptive mixed finite element methods for the stokes problem, arXiv:1403.0895 (2014).
I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
Nicholas J. Higham, Accuracy and stability of numerical algorithms, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR 1927606, DOI 10.1137/1.9780898718027
Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
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
Claes Johnson and J.-Claude Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. MR 583487, DOI 10.1090/S0025-5718-1980-0583487-9
Claes Johnson, Yi Yong Nie, and Vidar Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), no. 2, 277–291. MR 1043607, DOI 10.1137/0727019
Balázs Kovács, Buyang Li, and Christian Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal. 54 (2016), no. 6, 3600–3624. MR 3582825, DOI 10.1137/15M1040918
Christian Kreuzer and Kunibert G. Siebert, Decay rates of adaptive finite elements with Dörfler marking, Numer. Math. 117 (2011), no. 4, 679–716. MR 2776915, DOI 10.1007/s00211-010-0324-5
S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68. MR 270118, DOI 10.4064/sm-34-1-43-67
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
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
Tomasz Plewa, Timur Linde, and V. Gregory Weirs (eds.), Adaptive mesh refinement—theory and applications, Lecture Notes in Computational Science and Engineering, vol. 41, Springer, Berlin, 2005. MR 2789743, DOI 10.1007/b138538
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
Francisco-Javier Sayas, The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces, SIAM J. Numer. Anal. 47 (2009), no. 5, 3451–3463. MR 2551202, DOI 10.1137/08072334X
Christoph Schwab and Rob Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78 (2009), no. 267, 1293–1318. MR 2501051, DOI 10.1090/S0025-5718-08-02205-9
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
Gustaf Söderlind, Automatic control and adaptive time-stepping, Numer. Algorithms 31 (2002), no. 1-4, 281–310. Numerical methods for ordinary differential equations (Auckland, 2001). MR 1950926, DOI 10.1023/A:1021160023092
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
Tsogtgerel Gantumur, Adaptive boundary element methods with convergence rates, Numer. Math. 124 (2013), no. 3, 471–516. MR 3066037, DOI 10.1007/s00211-013-0524-x
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
Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. MR 1971217, DOI 10.1007/s002110100308