Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potentials with sufficient regularity the parameters of the algorithm can be designed such that the error of the approximated potential converges at any specified rate $O(h^p)$. The accuracy and effectiveness of the proposed algorithms are demonstrated for potentials of the Poisson equation in three dimensions.

翻译：提出了三种算法,用于评估在椭圆形PDE的边界要素方法中产生的体积潜力。 方法是对边界集中体积网块采用经过修改的快速多极方法。 如果美元是边界的网状体, 那么该体积将使用近O( h ⁇ -2}) 美元的自由度进行分离, 算法则计算出近O( h ⁇ -2}) 美元的复杂性。 这里几乎意味着可能会出现对数值$( h ⁇ -2} 美元) 。 因此, 体积潜力计算的复杂性与边界潜力相同。 对于有足够规律的源和潜力, 算法参数的设计可以使任何特定速率的近似潜在体积的误差达到$( h ⁇ -2} 美元。 拟议的算法的准确性和有效性表现为 Poisson 方程式三个维的潜能值 。