Triangular Elements in the Finite Element Method

Abstract

For a plane polygonal domain and a corresponding (general) triangulation we define classes of functions which are polynomials on each triangle and which are in and also belong to the Sobolev space . 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.