Orthogonal collocation on finite elements for elliptic equations

The method of orthogonal collocation on finite elements (OCFE) combines the features of orthogonal collocation with those of the finite element method. The method is illustrated for a Poisson equation (heat conduction with source term) in a rectangular domain. Two different basis functions are employed: either Hermite or Lagrange polynomials (with first derivative continuity imposed to ensure equivalence to the Hermite basis). Cubic or higher degree polynomials are used. The equations are solved using an LU-decomposition for the Hermite basis and an alternating direction implicit (ADI) method for the Lagrange basis.