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 −Δu = 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 comer. 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∞-error estimates in some weighted and unweighted norms also