Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P 1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H 1(Ω) and in the Neumann and Robin cases in L 2(Ω).
