Analysis of a Discontinuous Galerkin Method for Diffusion Problems on Intersecting Domains

The interior penalty discontinuous Galerkin method is applied to solve elliptic equations on either networks of segments or networks of planar surfaces, with arbitrary but fixed number of bifurcations. Stability is obtained by proving a discrete Poincar\'e's inequality on the hypergraphs. Convergence of the scheme is proved for $H^r$ regularity solution with $1 < r \leq 2$. In the low regularity case ($r \leq 3/2$), a weak consistency result is obtained via generalized lifting operators for Sobolev spaces defined on hypergraphs. Numerical experiments confirm the theoretical results.