## Interior maximum-norm estimates for finite element methods. II

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- 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, - 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, - 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

*Superconvergent finite element methods, Part*1,

*Construction of the methods and interior estimates*(to appear).

*Isolation of singularities of the Green’s function*, Math. USSR-Izv.

**1**(1967), 935-966.

## 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