Staggered DG method with small edges for Darcy flows in fractured porous media

TL;DR

This paper introduces a robust staggered discontinuous Galerkin method for Darcy flows in fractured porous media, capable of handling general meshes with small edges and anisotropic properties, with proven optimal convergence and confirmed by numerical experiments.

Contribution

The paper develops a new staggered DG method that weakens mesh assumptions and demonstrates optimal convergence, robustness, and applicability to unfitted meshes in fractured media.

Findings

Optimal $L^2$ error estimates for all variables.

Method remains robust with respect to heterogeneity and anisotropy.

Numerical experiments confirm theoretical results on complex meshes.

Abstract

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectively. Our current analysis weakens the usual assumption on the polygonal mesh, which can integrate more general meshes such as elements with arbitrarily small edges into our theoretical framework. We prove the optimal convergence estimates in $L^2$ error for all the variables by exploiting the Ritz projection. Importantly, our error estimates are shown to be fully robust with respect to the heterogeneity and anisotropy of the permeability coefficients. Several numerical experiments including meshes with small edges and anisotropic meshes are carried out to confirm the theoretical findings. Finally, our method is applied in the framework of unfitted mesh.