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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.