Mathematics of Computation

Some New Error Estimates for Ritz--Galerkin Methods with Minimal Regularity Assumptions

Author(s): Alfred H. Schatz; Junping Wang.
Journal: Math. Comp. 65 (1996), 19-27.
MSC (1991): Primary 65N30; Secondary 65F10
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Abstract: New uniform error estimates are established for finite element approximations $u_h$

of solutions $u$

of second-order elliptic equations $\mathcal L u = f$ using only the regularity assumption $\|u\|_1 \leq c\|f\|_{-1}$ . Using an Aubin--Nitsche type duality argument we show for example that, for arbitrary (fixed) $\varepsilon$ sufficiently small, there exists an $h_0$

such that for $0 < h < h_0$

$\begin{displaymath}\|u-u_h\|_0 \leq \varepsilon \|u-u_h\|_1. \end{displaymath}$

Here, $\|\cdot\|_s$ denotes the norm on the Sobolev space $H^s$

. Other related results are established.

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

Junping Wang
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071
Email: junping@schwarz.uwyo.edu

DOI: 10.1090/S0025-5718-96-00649-7
PII: S 0025-5718(96)00649-7
Received by editor(s): November 9, 1993
Additional Notes: This research was supported by NSF Grant DMS 9007185
Dedicated: Dedicated to Joachim Nitsche
Copyright of article: Copyright 1996, American Mathematical Society

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