  In this paper, we propose a novel adaptive finite element method for an
elliptic equation with line Dirac delta functions as a source term. We first
study the well-posedness and global regularity of the solution in the whole
domain. Instead of regularizing the singular source term and using the
classical residual-based a posteriori error estimator, we propose a novel a
posteriori estimator based on an equivalent transmission problem with zero
source term and nonzero flux jumps on line fractures. The transmission problem
is defined in the same domain as the original problem excluding on line
fractures, and the solution is therefore shown to be more regular. The
estimator relies on meshes conforming to the line fractures and its edge jump
residual essentially uses the flux jumps of the transmission problem on line
fractures. The error estimator is proven to be both reliable and efficient, an
adaptive finite element algorithm is proposed based on the error estimator and
the bisection refinement method. Numerical tests show that quasi-optimal
convergence rates are achieved even for high order approximations and the
adaptive meshes are only locally refined at singular points.
