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 Poincaré’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.

💡 Research Summary

This paper presents the formulation and rigorous numerical analysis of an interior penalty discontinuous Galerkin (DG) method for solving diffusion problems on complex, intersecting domains known as hypergraphs. These domains model real-world structures such as networks of one-dimensional segments (e.g., blood vessels) or two-dimensional planar surfaces (e.g., fracture networks in subsurface geology), where multiple elements meet at bifurcation points or lines.

The primary challenge addressed is that standard DG methods assume an interior facet is shared by exactly two elements, an assumption violated at the bifurcations of a hypergraph. The authors’ key innovation is a novel redefinition of the jump and average operators at these bifurcation nodes (Eq. 3.1). This generalized definition allows the Kirchhoff condition (flux balance) and solution continuity at bifurcations to be seamlessly incorporated into the DG formulation via a special flux term, as proven in Lemma 3.1.

The analysis begins by defining the model problem and its well-posed weak formulation on the hypergraph. The numerical scheme is then constructed on shape-regular meshes of the individual segments or surfaces. Stability of the method is established by proving a discrete Poincaré inequality on the hypergraph. The cornerstone of this proof is the construction of an enriching/averaging map, E_h (Theorem 3.2), which projects discontinuous DG functions onto a continuous subspace and provides crucial estimates linking the DG norm to standard Sobolev norms.

Convergence analysis is treated in two distinct regimes based on the regularity of the exact solution. For sufficiently smooth solutions (in H^r with 1 < r ≤ 2), optimal-order error estimates are derived using standard Galerkin orthogonality and projection estimates. The more significant and non-standard contribution lies in the low-regularity case (r ≤ 3/2). Here, Galerkin orthogonality fails. To overcome this, the authors develop a sophisticated functional framework, extending concepts from classical domains to hypergraphs. This includes defining generalized lifting operators and a weak notion of normal traces for Sobolev spaces on hypergraphs. Using these tools, they prove a weak consistency result for the bilinear form and subsequently derive a priori error bounds even in the absence of Galerkin orthogonality.

The theoretical developments are supported by numerical experiments on both edge-networks and plane-networks. These experiments confirm the predicted convergence rates for smooth solutions and demonstrate the method’s robustness and accuracy for solutions with low regularity.

In summary, this work provides a comprehensive and theoretically sound DG framework for elliptic problems on intersecting manifold networks. It successfully handles the complexities of bifurcations, proves stability via a discrete Poincaré inequality, and establishes convergence rates for both high and low solution regularities, the latter involving significant extensions of existing DG analysis tools to hypergraph domains.