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In a plane polygonal domain, consider a Poisson problem      −  Δ  u = f - \Delta u = f       with homogeneous Dirichlet boundary condition and the  p -version finite element solutions of this. We give various upper and lower bounds for the error measured in      L 2 {L^2}      . In the case of a single element (i.e., a convex domain), we reduce the question of sharpness of these estimates to the behavior of a certain inf-sup constant, which is numerically determined, and a likely sharp estimate is then conjectured. This is confirmed during a series of numerical experiments also for the case of a reentrant corner. For a one-dimensional analogue problem (of rotational symmetry), sharp      L 2 {L^2}      -error estimates are proven directly and via an extension of the classical duality argument. Here, we give sharp      L ∞  {L^\infty }      -error estimates in some weighted and unweighted norms also.

On the sharpness of 𝐿²-error estimates of 𝐻¹₀-projections onto subspaces of piecewise, high-order polynomials