We propose and analyze discontinuous Galerkin (dG) approximations to 3D-1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the time dependent problem, a backward Euler dG formulation is also presented and analysed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method.

翻译：我们提出并分析了用于三维-一维耦合系统的间断Galerkin（dG）逼近方法，该系统模拟了包含缩小至其一维中心线的小型夹杂物的三维域中的扩散。通过推导后验误差估计以及利用合适提升算子定义的残差界，我们建立了稳态问题弱解的收敛性。针对时间依赖问题，我们还提出并分析了向后欧拉dG公式。此外，我们提出了一种用于嵌入三维域中网络的dG方法，该方法在分岔点上（除跳跃项外）具有局部质量守恒性。理想化几何形状中的数值算例验证了我们的理论发现，而真实一维网络中的模拟则展示了我们方法的鲁棒性。