ISSN:
0945-3245
Keywords:
AMS(MOS): 65N30
;
CR: G.1.8
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary In this paper we settle a question concerning the necessity of the logh-factor appearing in error estimates of linear finite element solutions. Let Ω⊂ℝ2 be a bounded convex polygon and {T h } a regular family of subdivisions of Ω into triangles. Be furtherS 0 h ⊂H 0 1 (Ω) the corresponding linear finite element spaces andP h :H 0 1 (Ω)→S 0 h theH 0 1 -projections. With respect to the uniform norm asymptotically estimates of the form $$\parallel u - P_h u\parallel _\infty \leqq c|logh|dist(u,S_0^h ), u \in C(\bar \Omega ) \cap H_0^1 (\Omega )$$ hold, as pointed out by Schatz [1980]. For functionsu with bounded second derivatives this implies $$\parallel u - P_h u\parallel _\infty \leqq ch^2 |\log h||u|_{2,\infty } ,$$ a result which was proved by Nitsche [1975] and Scott [1976] with different methods. It was an open question if the factor |logh| is necessary or only due to the methods of proof. In this paper it is shown by an example that the cited estimates are sharp in the very strong sense that in general they are no longer valid if |logh| is replaced by a termo(|logh|).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01405570
