---
abstract: '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.'
address:
- 'Department of Mathematics, University of Bergen, Norway. '
- 'Department of Mathematics, Karlstad University, Sweden. '
- 'Department of Mathematics, Technical University of Munich, Germany. '
author:
- 'Ingeborg G. Gjerde'
- Kundan Kumar
- 'Jan M. Nordbotten'
- Barbara Wohlmuth
date: 'October 25, 2018'
title: Splitting method for elliptic equations with line sources
---

[^1]

Introduction
============

Decomposition and regularity properties
=======================================

Numerical methods
=================

Numerical results
=================

**Acknowledgements** The authors thank J. Reichenbach and A. Deistung for bringing our attention to the data used in section \[sec:num-brain\] [@brain], and E. Hanson and E. Hodneland for providing us with the data segmentation and tree extraction.

Conclusions
===========

We studied an elliptic equation having line sources in a 3D domain. The line sources act as Dirac measure defined on a line causing the solutions to be singular on the line itself. Central to this work is the result that the solution admits a split into a singular and a regular part. This allows us to study the nature of the solution as well as to develop a numerical algorithm for solving the problem. Mathematically, we see that the solution has anisotropic regularity, it is smooth along the line source and the line singularity acts as a Dirac point measure in a 2D domain. Our numerical approach solves for the regular part only and therefore obtains optimal convergence rates. We illustrate our approach for several numerical examples including a data set describing the vascular system of a human brain. Our solution approach is mesh-independent and can be adapted to a variety of discretizations. \[sec:conclusions\]

[^1]: This work was partially supported by the Research Council of Norway, project number 250223, and Deutsche Forschungsgemeinschaft, grant number WO-671/11-1.
