Discontinuous Galerkin methods for 3D-1D systems (Q6583660)

The authors propose and analyze interior penalty 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. Due to low regularity of weak solutions the method does not satisfy strong consistency. Thus, 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. This also leads to error estimates for regular and graded meshes.\N\NIn addition, a backward Euler dG formulation is presented and analyzed for the time-dependent (parabolic) problem. In particular, a priori error estimates are proven. Further, the authors propose a dG method for networks embedded in 3D domains using hybridization at bifurcation points. The method is, up to jump terms, locally mass conservative on bifurcation points.\N\NNumerical examples in idealized geometries complement the theoretical findings, and simulations in realistic 1D networks show the robustness of the method.