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

TL;DR

This paper analyzes a discontinuous Galerkin method for solving elliptic diffusion problems on intersecting networks, establishing stability, convergence, and weak consistency, supported by numerical experiments.

Contribution

It introduces a stability and convergence analysis of a DG method on hypergraph domains with arbitrary bifurcations, including low regularity cases.

Findings

Proved stability via a discrete Poincaré inequality on hypergraphs.

Established convergence for solutions with $H^r$ regularity, $1<r extless2$.

Validated theoretical results with numerical experiments.

Abstract

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.