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Recent results of Schatz show that standard Galerkin finite el- ement methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are local- ized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error. These results do not indicate that such localization occurs when piecewise linear elements are used. We show via simple one-dimensional examples that Schatz's estimates are sharp in that localization indeed does not occur when piecewise linear elements are used. @ @xj (aij(x) @u @xi) + PN i=1bi(x) @u @xi +c(x)u = f in , @u @nL = 0 on @ , where @u @nL is the conormal derivative and is a domain in R N with smooth bound- ary @. While the pointwise dependence of the error in the finite element method upon u and its derivatives is in general global, these weighted estimates show that, for all finite element spaces of polynomial order two and higher, standard Galerkin finite element methods are localized in the sense that the global dependence of the pointwise error occurs only via higher order terms. However, Schatz's results in- dicate (but do not prove) that no localization occurs in the lowest-order piecewise linear case. The purpose of this note is to investigate the sharpness of this result. Asymptotic error expansion inequalities were given in (Sch98) as elementary con- sequences of the weighted maximum norm estimates proved therein. These expan- sions are convenient for use in applications (cf. (HSWW01)) and display localization properties more clearly than do the weighted maximum norm estimates they are derived from, so we shall only state the expansions. We also note that sharpness of the expansions implies sharpness of the weighted norm results since the former are derived from the latter. We letuh 2 Sr h be the Galerkin approximation to a solution u to (1.1). Here r is the order of approximation given by Sr

Piecewise linear finite element methods are not localized