Interior maximum-norm estimates for finite element methods. II
HTML articles powered by AMS MathViewer
- by A. H. Schatz and L. B. Wahlbin PDF
- Math. Comp. 64 (1995), 907-928 Request permission
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
- James H. Bramble, Joachim A. Nitsche, and Alfred H. Schatz, Maximum-norm interior estimates for Ritz-Galerkin methods, Math. Comput. 29 (1975), 677–688. MR 0398120, DOI 10.1090/S0025-5718-1975-0398120-7 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).
- Chuan Miao Chen, $W^{1,\infty }$-interior estimates for finite element method on regular mesh, J. Comput. Math. 3 (1985), no. 1, 1–7. MR 815405
- P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Richard Haverkamp, Eine Aussage zur $L_{\infty }$-Stabilität und zur genauen Konvergenzordnung der $H^{1}_{0}$-Projektionen, Numer. Math. 44 (1984), no. 3, 393–405 (German, with English summary). MR 757494, DOI 10.1007/BF01405570 J. P. Krasovskiĭ, Isolation of singularities of the Green’s function, Math. USSR-Izv. 1 (1967), 935-966.
- 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
- 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
- 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, On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces, Math. Comp. 38 (1982), no. 157, 1–22. MR 637283, DOI 10.1090/S0025-5718-1982-0637283-6
- A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), no. 2, 505–521. MR 1388486, DOI 10.1137/0733027
- Martin Schechter, On $L^{p}$ estimates and regularity. I, Amer. J. Math. 85 (1963), 1–13. MR 188615, DOI 10.2307/2373179
Additional Information
- © Copyright 1995 American Mathematical Society
- 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