Abstract: We present a technique for numerically solving Dirichlet boundary-value problems on a general domain. We do not assume the domain polygonal. This is achieved by using suitably defined extensions from polyhedral subdomains; the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals.
The technique is independent of the representation of the boundary and of the space dimension. Moreover, it allows the use of polyhedral elements and high order approximations. In the polyhedral subdomains, we use a hybridizable discontinuous Galerkin method. We apply this technique to pure-diffusion, convection-diffusion and exterior diffusion problems and provide numerical experiments showing that the convergence properties of the resulting method are the same as those for the case when the domain is polygonal, whenever the distance between the boundary and the computational boundary is of order of the meshsize.