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I've got an elliptic problem in a domain with two materials in a checkerboard arrangement. Both materials are highly dissimilar, being one much more compliant than the other.

If I use classic linear lagrange finite element, because of the corners of the checkerboard, there will be singularities that will make the solution gradient not converge with the mesh refinement. How can I prove this fact? I am looking for a proof based on the fact that the material properties uses the same mesh discretization, but it's piecewise constant. Which references would be good for this proof? Thanks.

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