  We present a numerical scheme for the solution of nonlinear mixed-dimensional
PDEs describing coupled processes in embedded tubular network system in
exchange with a bulk domain. Such problems arise in various biological and
technical applications such as in the modeling of root-water uptake, heat
exchangers, or geothermal wells. The nonlinearity appears in form of
solution-dependent parameters such as pressure-dependent permeability or
temperature-dependent thermal conductivity. We derive and analyse a numerical
scheme based on distributing the bulk-network coupling source term by a
smoothing kernel with local support. By the use of local analytical solutions,
interface unknowns and fluxes at the bulk-network interface can be accurately
reconstructed from coarsely resolved numerical solutions in the bulk domain.
Numerical examples give confidence in the robustness of the method and show the
results in comparison to previously published methods. The new method
outperforms these existing methods in accuracy and efficiency. In a root water
uptake scenario, we accurately estimate the transpiration rate using only a few
thousand 3D mesh cells and a structured cube grid whereas other
state-of-the-art numerical schemes require millions of cells and local grid
refinement to reach comparable accuracy.

Computational Engineering, Finance, and Science