Several polynomial finite elements of nodal type are introduced that should lead to convergence of O(h1) in the L2 norm. Two of these methods are new and are expected to achieve the same orders of convergence with fewer parameters than the third method. They are applied to the one-group diffusion equation under different formulations, namely, several versions (with or without reduced and transverse integrations) of the primal and the mixed-hybrid formulations. Convergence rates are checked for a model problem with an analytical solution. Two of these methods exhibit superconvergence phenomena [O(h4) instead of O(h3)], a fact that can be explained heuristically. The most promising method, with only five parameters per cell, turns out to yield only O(h2) in its most algebraically efficient versions, while it has the potential of O(h3) convergence rates. Again, an explanation is given for this behavior and a fully O(h3) version is developed. Finally, these methods are applied to more realistic multigroup situations. In all cases, they are compared with results obtained from polynomial nodal methods in response matrix formalism. In the multigroup case, a well-known reference solution is also used.