Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Interior maximum-norm estimates for finite element methods. II


Authors: A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 64 (1995), 907-928
MSC: Primary 65N30; Secondary 65N15
DOI: https://doi.org/10.1090/S0025-5718-1995-1297478-7
MathSciNet review: 1297478
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider bilinear forms $ A( \bullet , \bullet )$ connected with second-order elliptic problems and assume that for $ {u_h}$ in a finite element space $ {S_h}$, we have $ A(u - {u_h},\chi ) = F(\chi )$ for $ \chi $ in $ {S_h}$ with local compact support. We give local estimates for $ u - {u_h}$ in $ {L_\infty }$ and $ W_\infty ^1$ of the type "local best approximation plus weak outside influences plus the local size of F".


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

  • [1] J. H. Bramble, J. A. Nitsche, and A. H. Schatz, Maximum norm interior estimates for Ritz-Galerkin methods, Math. Comp. 29 (1975), 677-688. MR 0398120 (53:1975)
  • [2] M.-E. Cayco, A. H. Schatz, and L. B. Wahlbin, Superconvergent finite element methods, Part 1, Construction of the methods and interior estimates (to appear).
  • [3] C.-M. Chen, $ {W^{1,\infty }}$-interior estimates for finite element method on regular mesh, J. Comput. Math. 3 (1985), 1-7. MR 815405 (87b:65194)
  • [4] P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, ed.), Vol. II, Part 1, North-Holland, Amsterdam, 1991, pp. 18-351. MR 1115237
  • [5] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, 1983. MR 737190 (86c:35035)
  • [6] R. Haverkamp, Eine Aussage zur $ {L_\infty }$-Stabilität und zur genauen Konvergenzordnung der $ H_0^1$-Projektionen, Numer. Math. 44 (1984), 393-405. MR 757494 (86f:65194)
  • [7] J. P. Krasovskiĭ, Isolation of singularities of the Green's function, Math. USSR-Izv. 1 (1967), 935-966.
  • [8] J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937-958. MR 0373325 (51:9525)
  • [9] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 437-445. MR 645661 (83e:65180)
  • [10] A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), 414-442. MR 0431753 (55:4748)
  • [11] -, On the quasi-optimality in $ {L_\infty }$ of the $ \ring{H}^1$-projection into finite element spaces, Math. Comp. 38 (1982), 1-21. MR 637283 (82m:65106)
  • [12] A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes which are locally symmetric with respect to a point, SIAM J. Numer. Anal. (to appear). MR 1388486 (98f:65112)
  • [13] M. Schechter, On $ {L^p}$ estimates and regularity I, Amer. J. Math. 85 (1963), 1-13. MR 0188615 (32:6051)

Similar Articles

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

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


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1297478-7
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society