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

Finite element method with local damage on the mesh