  Motivated by problems where the response is needed at select localized
regions in a large computational domain, we devise a novel finite element
discretization that results in exponential convergence at pre-selected points.
The two key features of the discretization are (a) use of midpoint integration
to evaluate the contribution matrices, and (b) an unconventional bending of the
mesh into complex space. Named complex-length finite element method (CFEM), the
technique is linked to Pade approximants that provide exponential convergence
of the Dirichlet-to-Neumann maps and thus the solution at specified points in
the domain. Exponential convergence facilitates drastic reduction in the number
of elements. This, combined with sparse computation associated with linear
finite elements, results in significant reduction in the computational cost.
The paper presents the basic ideas of the method as well as illustration of its
effectiveness for a variety of problems involving Laplace, Helmholtz and
elastodynamic equations.
