Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Dahmen, W.; Faermann, B.; Graham, I.; Hackbusch, W.; Sauter, S.

doi:10.1090/S0025-5718-03-01583-7

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method
HTML articles powered by AMS MathViewer

by W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch and S. A. Sauter;

Math. Comp. 73 (2004), 1107-1138

DOI: https://doi.org/10.1090/S0025-5718-03-01583-7

Published electronically: July 14, 2003

PDF | Request permission

Abstract:

We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions $u$ defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form $\left \|h^{\alpha }u\right \| _{W^{s,p}(\Omega )}$ for positive and negative $s$ and $\alpha$, where $h$ is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is $N$, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results—previously known only for quasi-uniform meshes—to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

References

Randolph E. Bank and L. Ridgway Scott, On the conditioning of finite element equations with highly refined meshes, SIAM J. Numer. Anal. 26 (1989), no. 6, 1383–1394. MR 1025094, DOI 10.1137/0726080
C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI (Paris, 1989–1991) Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13–51. MR 1268898
P. Binev, W. Dahmen, R. DeVore and P. Petrushev: Approximation classes for adaptive methods, Serdica Math. J., 28 (2002), 1001–1026.
Dietrich Braess, Finite elements, Cambridge University Press, Cambridge, 1997. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German original by Larry L. Schumaker. MR 1463151, DOI 10.1007/978-3-662-07233-2
D. Braess and W. Dahmen, Stability estimates of the mortar finite element method for $3$-dimensional problems, East-West J. Numer. Math. 6 (1998), no. 4, 249–263. MR 1667306
Dietrich Braess, Wolfgang Dahmen, and Christian Wieners, A multigrid algorithm for the mortar finite element method, SIAM J. Numer. Anal. 37 (1999), no. 1, 48–69. MR 1721260, DOI 10.1137/S0036142998335431
D. Braess, M. Dryja, and W. Hackbusch, A multigrid method for nonconforming FE-discretisations with application to non-matching grids, Computing 63 (1999), no. 1, 1–25. MR 1702163, DOI 10.1007/s006070050048
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
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
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Stephan Dahlke and Ronald A. DeVore, Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations 22 (1997), no. 1-2, 1–16. MR 1434135, DOI 10.1080/03605309708821252
W. Dahmen, B. Faermann, I.G. Graham, W. Hackbusch, S.A. Sauter, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Report 24/2001, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2001.
Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
Ronald A. DeVore and Vasil A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), no. 1, 397–414. MR 920166, DOI 10.1090/S0002-9947-1988-0920166-3
Ronald A. DeVore and Xiang Ming Yu, Degree of adaptive approximation, Math. Comp. 55 (1990), no. 192, 625–635. MR 1035930, DOI 10.1090/S0025-5718-1990-1035930-5
Birgit Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case, IMA J. Numer. Anal. 20 (2000), no. 2, 203–234. MR 1752263, DOI 10.1093/imanum/20.2.203
B. Faermann: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. Part II. The three-dimensional case. Numer. Math., 92 (2002), 467-499.
I. G. Graham, W. Hackbusch, and S. A. Sauter, Discrete boundary element methods on general meshes in 3D, Numer. Math. 86 (2000), no. 1, 103–137. MR 1774011, DOI 10.1007/PL00005399
I. G. Graham, W. Hackbusch, and S. A. Sauter, Hybrid Galerkin boundary elements: theory and implementation, Numer. Math. 86 (2000), no. 1, 139–172. MR 1774012, DOI 10.1007/PL00005400
W. Hackbusch, Elliptic differential equations, Springer Series in Computational Mathematics, vol. 18, Springer-Verlag, Berlin, 1992. Theory and numerical treatment; Translated from the author’s revision of the 1986 German original by Regine Fadiman and Patrick D. F. Ion. MR 1197118, DOI 10.1007/978-3-642-11490-8
Kenneth Eriksson, Don Estep, Peter Hansbo, and Claes Johnson, Introduction to adaptive methods for differential equations, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 105–158. MR 1352472, DOI 10.1017/S0962492900002531
Chisup Kim, Raytcho D. Lazarov, Joseph E. Pasciak, and Panayot S. Vassilevski, Multiplier spaces for the mortar finite element method in three dimensions, SIAM J. Numer. Anal. 39 (2001), no. 2, 519–538. MR 1860265, DOI 10.1137/S0036142900367065
Hans Triebel, Interpolation theory, function spaces, differential operators, 2nd ed., Johann Ambrosius Barth, Heidelberg, 1995. MR 1328645
Albert H. Schatz, Vidar Thomée, and Wolfgang L. Wendland, Mathematical theory of finite and boundary element methods, DMV Seminar, vol. 15, Birkhäuser Verlag, Basel, 1990. MR 1116555, DOI 10.1007/978-3-0348-7630-8
Barbara I. Wohlmuth, Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers, SIAM J. Numer. Anal. 36 (1999), no. 5, 1636–1658. MR 1710465, DOI 10.1137/S0036142997330512
B. Wohlmuth, Discretization methods and iterative solvers based on domain decomposition, Habilitation Thesis, University of Augsburg, November 1999.