  The accurate and efficient evaluation of potentials is of great importance
for the numerical solution of partial differential equations. When the
integration domain of the potential is irregular and is discretized by an
unstructured mesh, the function spaces of near field and self-interactions are
non-compact, and, thus, their computations cannot be easily accelerated. In
this paper, we propose three novel and complementary techniques for
accelerating the evaluation of potentials over unstructured meshes. Firstly, we
rigorously characterize the geometry of the near field, and show that this
analysis can be used to eliminate all the unnecessary near field interaction
computations. Secondly, as the near field can be made arbitrarily small by
increasing the order of the far field quadrature rule, the expensive near field
interaction computation can be efficiently offloaded onto the FMM-based far
field interaction computation, which leverages the computational efficiency of
highly optimized parallel FMM libraries. Finally, we show that a separate
interpolation mesh that is staggered to the quadrature mesh dramatically
reduces the cost of constructing the interpolants. Besides these contributions,
we present a robust and extensible framework for the evaluation and
interpolation of 2-D volume potentials over complicated geometries. We
demonstrate the effectiveness of the techniques with several numerical
experiments.
