  We study the elliptic equation with a line Dirac delta function as the source
term subject to the Dirichlet boundary condition in a two-dimensional domain.
Such a line Dirac measure causes different types of solution singularities in
the neighborhood of the line fracture. We establish new regularity results for
the solution in a class of weighted Sobolev spaces and propose finite element
algorithms that approximate the singular solution at the optimal convergence
rate. Numerical tests are presented to justify the theoretical findings.
