This may not be what you seek, but in the 1996 paper,
"Anisotropic refinement algorithms for finite elements"
by Goodman, Samuelsson, and Szepessy (.ps link), they
show an example of a function $w(x,y)=\frac{1}{2} u(x,y)=\frac{1}{2} y^2$, independent of $x$,
which solves $\Delta w u = 1$ on $\mathbb{R}^2$.
But with the triangulation shown below, with $\delta \ll \epsilon$, as $\delta \rightarrow 0$,
the finite element equation approximates $\Delta w u = 0$ instead of $\Delta w u = 1$.
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This may not be what you seek, but in the 1996 paper,
"Anisotropic refinement algorithms for finite elements"
by Goodman, Samuelsson, and Szepessy (.ps link), they
show an example of a function $w(x,y)=\frac{1}{2} y^2$, independent of $x$,
which solves $\Delta w = 1$ on $\mathbb{R}^2$.
But with the triangulation shown below, with $\delta \ll \epsilon$, as $\delta \rightarrow 0$,
the finite element equation approximates $\Delta w = 0$ instead of $\Delta w = 1$.
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This may not be what you seek, but in the 1996 paper,
"Anisotropic refinement algorithms for finite elements"
by Goodman, Samuelsson, and Szepessy (.ps link), they
show an example of a function $w(x,y)=\frac{1}{2} y^2$, independent of $x$,
which solves $\Delta w = 1$ on $\mathbb{R}^2$.
But with the triangulation shown below, as $\delta \rightarrow 0$,
the finite element equation approximates $\Delta w = 0$ instead of $\Delta w = 1$.
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