  We consider the finite element method on locally damaged meshes allowing for
some distorted cells which are isolated from one another. In the case of the
Poisson equation and piecewise linear Lagrange finite elements, we show that
the usual a priori error estimates remain valid on such meshes. We also propose
an alternative finite element scheme which is optimally convergent and,
moreover, well conditioned, i.e. its conditioning is of the same order as that
of a standard finite element method on a regular mesh of comparable size.
