Best approximation property in the 𝑊¹_{∞} norm for finite element methods on graded meshes

Demlow, A.; Leykekhman, D.; Schatz, A.; Wahlbin, L.

doi:10.1090/S0025-5718-2011-02546-9

Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes
HTML articles powered by AMS MathViewer

by A. Demlow, D. Leykekhman, A. H. Schatz and L. B. Wahlbin PDF

Math. Comp. 81 (2012), 743-764 Request permission

Abstract:

We consider finite element methods for a model second-order elliptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the $W^1_{\infty }$ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof technique. This result holds under a condition on the grid which is mildly more restrictive than the shape regularity condition typically enforced in adaptive codes. The second main contribution of this work is a discussion of the properties of and relationships between similar mesh restrictions that have appeared in the literature.

References

Thomas Apel, Arnd Rösch, and Dieter Sirch, $L^\infty$-error estimates on graded meshes with application to optimal control, SIAM J. Control Optim. 48 (2009), no. 3, 1771–1796. MR 2516188, DOI 10.1137/080731724
I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math. 34 (1980), no. 1, 41–62. MR 560793, DOI 10.1007/BF01463997
Nikolai Yu. Bakaev, Michel Crouzeix, and Vidar Thomée, Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations, M2AN Math. Model. Numer. Anal. 40 (2006), no. 5, 923–937 (2007). MR 2293252, DOI 10.1051/m2an:2006040
Nikolai Yu. Bakaev, Vidar Thomée, and Lars B. Wahlbin, Maximum-norm estimates for resolvents of elliptic finite element operators, Math. Comp. 72 (2003), no. 244, 1597–1610. MR 1986795, DOI 10.1090/S0025-5718-02-01488-6
James H. Bramble, Joseph E. Pasciak, and Olaf Steinbach, On the stability of the $L^2$ projection in $H^1(\Omega )$, Math. Comp. 71 (2002), no. 237, 147–156. MR 1862992, DOI 10.1090/S0025-5718-01-01314-X
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
Carsten Carstensen, Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for $H^1$-stability of the $L^2$-projection onto finite element spaces, Math. Comp. 71 (2002), no. 237, 157–163. MR 1862993, DOI 10.1090/S0025-5718-01-01316-3
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
M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
E. Dari, R. G. Durán, and C. Padra, Maximum norm error estimators for three-dimensional elliptic problems, SIAM J. Numer. Anal. 37 (2000), no. 2, 683–700. MR 1740762, DOI 10.1137/S0036142998340253
Klaus Deckelnick and Michael Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal. 45 (2007), no. 5, 1937–1953. MR 2346365, DOI 10.1137/060652361
Alan Demlow, Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quadilinear elliptic problems, SIAM J. Numer. Anal. 44 (2006), no. 2, 494–514. MR 2218957, DOI 10.1137/040610064
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, Sharply localized pointwise and $W_\infty ^{-1}$ estimates for finite element methods for quasilinear problems, Math. Comp., 76 (2007), pp. 1725–1741 (electronic).
Alan Demlow, Johnny Guzmán, and Alfred H. Schatz, Local energy estimates for the finite element method on sharply varying grids, Math. Comp. 80 (2011), no. 273, 1–9. MR 2728969, DOI 10.1090/S0025-5718-2010-02353-1
Alan Demlow and Rob Stevenson, Convergence and quasi-optimality of an adaptive finite element method for controlling $L_2$ errors, Numer. Math. 117 (2011), no. 2, 185–218. MR 2754849, DOI 10.1007/s00211-010-0349-9
Jim Douglas Jr. and Todd Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), 689–696. MR 431747, DOI 10.1090/S0025-5718-1975-0431747-2
R. G. Durán, An elementary proof of the continuity from $L^2_0(\Omega )$ to $H^1_0(\Omega )^n$ of Bogovskii’s right inverse of the divergence, preprint.
Kenneth Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Math. Models Methods Appl. Sci. 4 (1994), no. 3, 313–329. MR 1282238, DOI 10.1142/S0218202594000194
Stephen J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), no. 1, 225–233. MR 1156467, DOI 10.1090/S0002-9939-1993-1156467-3
Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205, DOI 10.1007/978-1-4612-5364-8
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
J. Guzmán, D. Leykekhman, J. Rossmann, and A. H. Schatz, Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods, Numer. Math. 112 (2009), no. 2, 221–243. MR 2495783, DOI 10.1007/s00211-009-0213-y
Yinnian He, Jinchao Xu, Aihui Zhou, and Jian Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), no. 3, 415–434. MR 2399151, DOI 10.1007/s00211-008-0141-2
Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
Ricardo H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp. 64 (1995), no. 209, 1–22. MR 1270622, DOI 10.1090/S0025-5718-1995-1270622-3
R. H. Nochetto, M. Paolini, and C. Verdi, An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates, Math. Comp. 57 (1991), no. 195, 73–108, S1–S11. MR 1079028, DOI 10.1090/S0025-5718-1991-1079028-X
Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
Patricia Saavedra and L. Ridgway Scott, Variational formulation of a model free-boundary problem, Math. Comp. 57 (1991), no. 196, 451–475. MR 1094958, DOI 10.1090/S0025-5718-1991-1094958-0
Alfred H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp. 67 (1998), no. 223, 877–899. MR 1464148, DOI 10.1090/S0025-5718-98-00959-4
Alfred H. Schatz, Maximum norm error estimates for the finite element method allowing highly refined grids, Recent advances in adaptive computation, Contemp. Math., vol. 383, Amer. Math. Soc., Providence, RI, 2005, pp. 133–139. MR 2195798, DOI 10.1090/conm/383/07161
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI 10.1090/S0025-5718-1977-0431753-X
A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI 10.1090/S0025-5718-1978-0502065-1
A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements, Math. Comp. 33 (1979), no. 146, 465–492. MR 502067, DOI 10.1090/S0025-5718-1979-0502067-6
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI 10.1090/S0025-5718-1995-1297478-7
Reinhard Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numér. 12 (1978), no. 1, 85–90, iii (English, with French summary). MR 483557, DOI 10.1051/m2an/1978120100851
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
Lars B. Wahlbin, Local behavior in finite element methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR 1115238
Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000), no. 231, 881–909. MR 1654026, DOI 10.1090/S0025-5718-99-01149-7