Splitting method for elliptic equations with line sources

¹ Department of Mathematics, University of Bergen, Bergen, Norway
² Department of Mathematics, Karlstad University, Karlstad, Sweden
³ Department of Mathematics, Technical University of Munich, Garching, Germany

^* Corresponding author: ingeborg.gjerde@uib.no

Received: 25 October 2018
Accepted: 18 April 2019

Abstract

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 its correction function w. This recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to L², a problem for which the discretizations and solvers are readily available. Moreover, as w enjoys higher regularity than the full solution, this 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 ~3000 line segments) describing the vascular system of the brain.