Numerical Investigation of Convergence Rates for the FEM Approximation of 3D-1D Coupled Problems

Abstract

We consider the numerical approximation of second order elliptic equations with singular forcing terms. In particular we investigate the case where a Dirac measure on a one-dimensional (1D) manifold is the forcing term for a three-dimensional (3D) problem. A partial differential equation is also defined on the manifold. The two problems are coupled by means of the intensity of the Dirac measure, which depends on both solutions. Such a problem is used to model the interaction of microcirculation and interstitial flow at the microscale, where the complicated geometrical configuration of the capillary network is taken into account. In order to facilitate the numerical discretization, the capillary bed is modeled as a collection of connected one-dimensional manifolds able to carry blood flow. We apply the finite element method (FEM) to discretize the equations in the interstitial volume and the capillary network. Because of the singular forcing terms, the solution of the coupled problem is not regular enough to apply the standard error analysis. A novel theoretical framework has been recently proposed to analyze elliptic problems with Dirac right hand sides. Using numerical experiments, in this work we investigate the validity of the available error estimates in the more general case of 3D-1D coupled problems, where the 1D problem acts as a concentrated source embedded in the surrounding volume.