The finite element method for elliptic boundary value problems has been modified to deal with boundary singularities. We introduce a singular-super-element (SSE) which incorporates the known expansion for the singular solution explicitly over the internal region surrounding the singular point, whilst using blended trial functions over the intermediate region, which joins the internal and external regions smoothly. The SSE conforms with the mesh used in the external region, and may be easily incorporated into standard finite element programs. The calculations yield the expansion coefficients directly, as well as an accurate representation of the displacements in the vicinity of the singular point, for a crack or V-notch of any angle subject to any mode of loading. The SSE has been applied to determine stress intensity factors for two-dimensional crack and V-notch problems, including mixed mode. The computations converge rapidly, yielding results of high accuracy.