  In this paper, we study the mathematical structure and numerical
approximation of elliptic problems posed in a (3D) domain $\Omega$ when the
right-hand side is a (1D) line source $\Lambda$. The analysis and approximation
of such problems is known to be non-standard as the line source causes the
solution to be singular. Our main result is a splitting theorem for the
solution; we show that the solution admits a split into an explicit, low
regularity term capturing the singularity, and a high-regularity correction
term $w$ being the solution of a suitable elliptic equation. The splitting
theorem states the mathematical structure of the solution; in particular, we
find that the solution has anisotropic regularity. More precisely, the solution
fails to belong to $H^1$ in the neighbourhood of $\Lambda$, but exhibits
piecewise $H^2$-regularity parallel to $\Lambda$. The splitting theorem can
further be used to formulate a numerical method in which the solution is
approximated via its correction function $w$. This approach has several
benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D
right-hand side belonging to $L^2$, a problem for which the discretizations and
solvers are readily available. Secondly, it makes the numerical approximation
independent of the discretization of $\Lambda$; thirdly, it improves the
approximation properties of the numerical method. We consider here the Galerkin
finite element method, and show that the singularity subtraction then recovers
optimal convergence rates on uniform meshes, i.e., without needing to refine
the mesh around each line segment. The numerical method presented in this paper
is therefore well-suited for applications involving a large number of line
segments. We illustrate this by treating a dataset (consisting of $\sim 3000$
line segments) describing the vascular system of the brain.
