  The large sparse linear systems arising from the finite element or finite
difference discretization of elliptic PDEs can be solved directly via, e.g.,
nested dissection or multifrontal methods. Such techniques reorder the nodes in
the grid to reduce the asymptotic complexity of Gaussian elimination from
$O(N^{2})$ to $O(N^{1.5})$ for typical problems in two dimensions. It has
recently been demonstrated that the complexity can be further reduced to O(N)
by exploiting structure in the dense matrices that arise in such computations
(using, e.g., $\mathcal{H}$-matrix arithmetic). This paper demonstrates that
such \textit{accelerated} nested dissection techniques become particularly
effective for boundary value problems without body loads when the solution is
sought for several different sets of boundary data, and the solution is
required only near the boundary (as happens, e.g., in the computational
modeling of scattering problems, or in engineering design of linearly elastic
solids.
