  Hollow-core holey fibers are promising candidates for low-loss guidance of
light in various applications, e.g., for the use in laser guide star adaptive
optics systems in optical astronomy. We present an accurate and fast method for
the computation of light modes in arbitrarily shaped waveguides. Maxwell's
equations are discretized using vectorial finite elements (FEM). We discuss how
we utilize concepts like adaptive grid refinement, higher-order finite
elements, and transparent boundary conditions for the computation of leaky
modes in photonic crystal fibers. Further, we investigate the convergence
behavior of our methods. We employ our FEM solver to design hollow-core
photonic crystal fibers (HCPCF) whose cores are formed from 19 omitted cladding
unit cells. We optimize the fiber geometry for minimal attenuation using
multidimensional optimization taking into account radiation loss (leaky modes).
