Splitting method for elliptic equations with line sources

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.

WITHDRAWN: Modeling tissue perfusion in terms of 1d-3d embedded mixed-dimension coupled problems with distributed sources

“…However, due to global interaction of those contributions, the numerical solution of the point source strengths results in dense system matrices. In [30], the support of the line source contributions is narrowed by a smooth cut-off function and the correction function is numerically approximated, yielding sparse system matrices. This method is also known as subtraction method from the field of electroencephalography (EEG) source reconstruction [31].…”

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

“…Discussion of other discretization schemes can be found in [7], again with the restriction of two-dimensional grids. Enhancements include, among others, three-dimensional slanted wells [8,9], Green's functions for the computation of well indices [10,11,12], or the singularity subtraction method to obtain smooth solutions in the near-well region [13,14].…”

A singularity removal method for coupled 1D–3D flow models

“…To see that the constructed p indeed solves the right problem, let us start by inserting it into (20). construction, all terms disappear except E(f )Ψ ∆G.…”

“…We are, to the best of our knowledge, the first to formulate a singularity removal method for the coupled 1D-3D flow problem. Central to this method is the construction of a function G w capturing the solution singularity; we use here a function G w found by integrating the Green's function for the reservoir equations (1a)-(1b) over the line Λ; we refer here to our earlier work in [20,Section 3.2]. This use of Green's functions to construct analytical and semi-analytical well models has a rich history.…”

“…As for the reservoir equation, the reservoir pressure could be replaced with a potential expression φ so that the effect of gravity can be included. The singularity removal and reformulation can be extended to handle spatially varying, scalarvalued permeabilities as shown in [20]. For an extension to tensor-valued permeabilities and non-linear reservoir equations, we suggest using the solution splitting in (6) to formulate a multiscale finite volume method such as in [40], or a generalized finite element method [37], where the analytic functions capturing the solution singularity are used to enrich the set of basis functions.…”