Classical potential theory converts linear constant-coefficient elliptic
problems in complex domains into integral equations on interfaces, and
generates robust, efficient numerical methods. The conversion is
usually carried out for a particular situation such as the Poisson
equation in dimension 2, and the efficiency of the resulting methods
then depends on detailed analysis of the appropriate special functions.

We present a general conversion scheme which leads naturally to a fast
general algorithm: arbitrary elliptic problems in arbitrary dimension
are converted to first-order systems, a periodic fundamental solution is
mollified for convergence, and the mollification is locally corrected
via Ewald summation. Local linear algebra and the elementary theory of
distributions yield a simple boundary integral equation. With the aid
of a new nonequidistant fast Fourier transform for piecewise polynomial
functions, the resulting numerical methods provide highly accurate
solutions to general elliptic systems in complex domains.