On the sharpness of $L^ 2$-error estimates of $H^ 1_ 0$-projections onto subspaces of piecewise, high-order polynomials

Abstract:

In a plane polygonal domain, consider a Poisson problem $- \Delta u = f$ 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 ${L^2}$. 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 ${L^2}$-error estimates are proven directly and via an extension of the classical duality argument. Here, we give sharp ${L^\infty }$-error estimates in some weighted and unweighted norms also.