  Problems with localized nonhomogeneous material properties arise frequently
in many applications and are a well-known source of difficulty in numerical
simulations. In certain applications (including additive manufacturing), the
physics of the problem may be considerably more complicated in relatively small
portions of the domain, requiring a significantly finer local mesh compared to
elsewhere in the domain. This can make the use of a uniform mesh numerically
unfeasible. While nonuniform meshes can be employed, they may be challenging to
generate (particularly for regions with complex boundaries) and more difficult
to precondition. The problem becomes even more prohibitive when the region
requiring a finer-level mesh changes in time, requiring the introduction of
refinement and derefinement techniques. To address the aforementioned
challenges, we employ a technique related to the Fat boundary method as a
possible alternative. We analyze the proposed methodology, from a mathematical
point of view and validate our findings on two-dimensional numerical tests.
