  We propose a method to reduce the computational effort to solve a partial
differential equation on a given domain. The main idea is to split the domain
of interest in two subdomains, and to use different approximation methods in
each of the two subdomains. In particular, in one subdomain we discretize the
governing equations by a canonical scheme, whereas in the other one we solve a
reduced order model of the original problem. Different approaches to couple the
low-order model to the usual discretization are presented. The effectiveness of
these approaches is tested on numerical examples pertinent to non-linear model
problems including the Laplace equation with non-linear boundary conditions and
the compressible Euler equations.
