  Electrostatic interactions in solvents play a major role in biophysical
systems. There is a consensus in the literature that the dielectric response of
aqueous solutions is nonlocal: polarization depends on the electric field not
only at a given point but in the vicinity of that point as well. This is
typically modeled via a convolution of the electric field with an appropriate
integral kernel.
  A primary problem with nonlocal models is high computational cost. A
secondary problem is restriction of convolution integrals to the solvent, as
opposed to their evaluation over the whole space.
  The paper develops a computational tool alleviating the "curse of
nonlocality" and helping to handle the integration correctly. This tool is
Trefftz approximations, which tend to furnish much higher accuracy than
traditional polynomial ones. In the paper, Trefftz approximations are developed
for problems of nonlocal electrostatics, with the goal of numerically
"localizing" the original nonlocal problem. This approach can be extended to
nonlocal problems in other areas of computational mathematics, physics and
engineering.
