  The diffuse domain method for partial differential equations on complicated
geometries recently received strong attention in particular from practitioners,
but many fundamental issues in the analysis are still widely open. In this
paper we study the diffuse domain method for approximating second order
elliptic boundary value problems posed on bounded domains, and show convergence
and rates of the approximations generated by the diffuse domain method to the
solution of the original second order problem when complemented by Robin,
Dirichlet or Neumann conditions. The main idea of the diffuse domain method is
to relax these boundary conditions by introducing a family of phase-field
functions such that the variational integrals of the original problem are
replaced by a weighted average of integrals of perturbed domains. From an
functional analytic point of view, the phase-field functions naturally lead to
weighted Sobolev spaces for which we present trace and embedding results as
well as various type of Poincar\'e inequalities with constants independent of
the domain perturbations. Our convergence analysis is carried out in such
spaces as well, but allows to draw conclusions also about unweighted norms
applied to restrictions on the original domain. Our convergence results are
supported by numerical examples.
