Piecewise linear finite element methods are not localized

Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized 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.