Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Best approximation property in the $ W^1_{\infty}$ norm for finite element methods on graded meshes


Authors: A. Demlow, D. Leykekhman, A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 81 (2012), 743-764
MSC (2010): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-2011-02546-9
Published electronically: September 29, 2011
MathSciNet review: 2869035
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. T. Apel, A. Rösch, and D. Sirch, $ L\sp \infty $-error estimates on graded meshes with application to optimal control, SIAM J. Control Optim., 48 (2009), pp. 1771-1796. MR 2516188 (2010f:49064)
  • 2. I. Babuška and J. Osborn, Analysis of finite element methods for second order boundary value problems using mesh dependent norms, Numer. Math., 34 (1980), pp. 41-62. MR 560793 (81g:65143)
  • 3. N. Y. Bakaev, M. Crouzeix, and V. Thomée, Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations, M2AN Math. Model. Numer. Anal., 40 (2006), pp. 923-937 (2007). MR 2293252 (2007k:35194)
  • 4. N. Y. Bakaev, V. Thomée, and L. B. Wahlbin, Maximum-norm estimates for resolvents of elliptic finite element operators, Math. Comp., 72 (2003), pp. 1597-1610 (electronic). MR 1986795 (2004g:65074)
  • 5. J. H. Bramble, J. E. Pasciak, and O. Steinbach, On the stability of the $ L^2$ projection in $ H^1(\Omega )$, Math. Comp., 71 (2002), pp. 147-156 (electronic). MR 1862992 (2002h:65175)
  • 6. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, Heidelberg, New York, third ed., 2008. MR 2373954 (2008m:65001)
  • 7. C. 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), pp. 157-163 (electronic). MR 1862993 (2002i:65125)
  • 8. J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524-2550. MR 2421046 (2009h:65174)
  • 9. M. Crouzeix and V. Thomée, The stability in $ L\sb p$ and $ W\sp 1\sb p$ of the $ L\sb 2$-projection onto finite element function spaces, Math. Comp., 48 (1987), pp. 521-532. MR 878688 (88f:41016)
  • 10. E. Dari, R. G. Durán, and C. Padra, Maximum norm error estimators for three-dimensional elliptic problems, SIAM J. Numer. Anal., 37 (2000), pp. 683-700 (electronic). MR 1740762 (2001b:65120)
  • 11. K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), pp. 1937-1953 (electronic). MR 2346365 (2008k:49005)
  • 12. A. 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), pp. 494-514 (electronic). MR 2218957 (2007c:65105)
  • 13. 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).
  • 14. A. Demlow, J. Guzmán, and A. Schatz, Local energy estimates for the finite element method on sharply varying grids, Math. Comp., 80 (2011), 1-9. MR 2728969
  • 15. A. Demlow and R. 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
  • 16. J. Douglas, Jr. and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., 29 (1975), pp. 689-696. MR 0431747 (55:4742)
  • 17. 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.
  • 18. K. Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Math. Models Methods Appl. Sci., 4 (1994), pp. 313-329. MR 1282238 (95c:65180)
  • 19. S. J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc., 119 (1993), pp. 225-233. MR 1156467 (93k:35076)
  • 20. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, Berlin, Heidelberg, New York, 1994. MR 1284205 (95i:35216a)
  • 21. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985. MR 775683 (86m:35044)
  • 22. M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), pp. 303-342. MR 657523 (83h:35033)
  • 23. 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), pp. 221-243. MR 2495783 (2010a:65237)
  • 24. Y. He, J. Xu, A. Zhou, and J. Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math., 109 (2008), pp. 415-434. MR 2399151 (2009j:76157)
  • 25. J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28 (1974), pp. 937-958. MR 0373325 (51:9525)
  • 26. R. H. Nochetto, Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp., 64 (1995), pp. 1-22. MR 1270622 (95c:65172)
  • 27. 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), pp. 73-108, S1-S11. MR 1079028 (92a:65322)
  • 28. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38 (1982), pp. 437-445. MR 645661 (83e:65180)
  • 29. P. Saavedra and L. R. Scott, Variational formulation of a model free-boundary problem, Math. Comp., 57 (1991), pp. 451-475. MR 1094958 (92a:35166)
  • 30. A. 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), pp. 877-899. MR 1464148 (98j:65082)
  • 31. height 2pt depth -1.6pt width 23pt, Maximum norm error estimates for the finite element method allowing highly refined grids, in Recent advances in adaptive computation, vol. 383 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2005, pp. 133-139. MR 2195798
  • 32. A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp., 31 (1977), pp. 414-442. MR 0431753 (55:4748)
  • 33. height 2pt depth -1.6pt width 23pt, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp., 32 (1978), pp. 73-109. MR 0502065 (58:19233a)
  • 34. height 2pt depth -1.6pt width 23pt, Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements, Math. Comp., 33 (1979), pp. 465-492. MR 0502067 (58:19233b)
  • 35. height 2pt depth -1.6pt width 23pt, Interior maximum-norm estimates for finite element methods. II, Math. Comp., 64 (1995), pp. 907-928. MR 1297478 (95j:65143)
  • 36. R. Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numér., 12 (1978), pp. 85-90, iii. MR 0483557 (58:3549)
  • 37. R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227-241 (electronic). MR 2353951 (2008j:65219)
  • 38. L. B. Wahlbin, Local behavior in finite element methods, in Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353-522. MR 1115238
  • 39. J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp., 69 (2000), pp. 881-909. MR 1654026 (2000j:65102)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N15

Retrieve articles in all journals with MSC (2010): 65N30, 65N15


Additional Information

A. Demlow
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: alan.demlow@uky.edu.

D. Leykekhman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: leykekhman@math.uconn.edu

A. H. Schatz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853.
Email: schatz@math.cornell.edu

L. B. Wahlbin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853.
Email: wahlbin@math.cornell.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02546-9
Keywords: Maximum norm, finite element, optimal error estimates
Received by editor(s): March 2, 2010
Received by editor(s) in revised form: October 27, 2010
Published electronically: September 29, 2011
Additional Notes: The first author was partially supported by NSF grant DMS-0713770.
The second author was partially supported by NSF grant DMS-0811167.
The third author was partially supported by NSF grant DMS-0612599.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society