The authors describe and analyze a method for computing inclusions on eigenvalues and eigenvectors of linear self-adjoint elliptic partial differential equations in two space dimensions. The requirement for precise inclusions forces the use of interval arithmetic. The method used requires the solution of a related Poisson equation with a variety of right-hand sides generated in a fixed point iteration for computing very accurate eigenvalues and eigenvectors. These solutions are computed using finite elements. The two computational examples are of eigenproblems generated by linearization of nonlinear elliptic problems: the Emden equation and the Allen-Cahn equation. The lowest eigenvalue is computed precisely, because it gives information about the existence of a solution of the nonlinear elliptic problems.