  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.
