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The paper analyses the convergence of the $p$-version of the FEM when thesolution is piecewise analytic function. It focuses on pointwise convergence ofthe gradient. It shows that at boundary the rate is different than inside theelement and there is a Gibbs phenomenon in the neighborhood of the point wherethe solution is not analytic. The major result is a conjecture written in theform of a theorem. The conjecture is based on careful numerical computations.The known theoretical results are stated.

On the $p$-version of FEM in one dimension: The known and unknown features