For a plane polygonal domain Ω \Omega and a corresponding (general) triangulation we define classes of functions p m ( x , y ) {p_m}(x,y) which are polynomials on each triangle and which are in C ( m ) ( Ω ) {C^{(m)}}(\Omega ) and also belong to the Sobolev space W 2 ( m + 1 ) ( Ω ) W_2^{(m + 1)}(\Omega ) . Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.