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In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain Ω when the right-hand side is a (1D) line source Λ. 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 H¹ in the neighbourhood of Λ, but exhibits piecewise H²-regularity parallel to Λ. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via …