  This article presents a fast solver for the dense "frontal" matrices that
arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The
solver relies on the fact that these matrices can be efficiently represented as
a hierarchically off-diagonal low-rank (HODLR) matrix. To construct the
low-rank approximation of the off-diagonal blocks, we propose a new
pseudo-skeleton scheme, the boundary distance low-rank approximation, that
picks rows and columns based on the location of their corresponding vertices in
the sparse matrix graph. We compare this new low-rank approximation method to
the adaptive cross approximation (ACA) algorithm and show that it achieves
betters speedup specially for unstructured meshes. Using the HODLR direct
solver as a preconditioner (with a low tolerance) to the GMRES iterative
scheme, we can reach machine accuracy much faster than a conventional LU
solver. Numerical benchmarks are provided for frontal matrices arising from 3D
finite element problems corresponding to a wide range of applications.
