The purpose of the present paper is to show how the results of the author [ibid. 21, 1180-1207 (1984; Zbl 0572.65074)] may be used to determine the approximability of some model problems in the usual Sobolev spaces by piecewise polynomials satisfying appropriate boundary and conformality conditions. Numerical results for two-dimensional linear elasticity are presented. The computations show that the predicted order of convergence is achieved even for low values of p. Some practical implications of the p-version convergence for the solvability of elliptic problems with strong singularities are also discussed.