  This paper shows how numerical methods on a regular grid in a box can be used
to generate numerical schemes for problems in general smooth domains contained
in the box with no need for a domain specific discretization. The focus is
mainly be on spectral discretizations due to their ability to accurately
resolve the interaction of finite order distributions (generalized functions)
and smooth functions. Mimicking the analytical structure of the relevant
(pseudodifferential) operators leads to viable and accurate numerical
representations and algorithms. An important byproduct of the structural
insights gained in the process is the introduction of smooth kernels (at the
discrete level) to replace classical singular kernels which are typically used
in the (numerical) representations of the solution. The new kernel
representations yield enhanced numerical resolution and, while they necessarily
lead to significantly higher condition numbers, they also suggest natural and
effective ways to precondition the systems.
