Using an Encompassing Periodic Box to Perform Numerical Calculations on General Domains

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.