Adaptive staggered DG method for Darcy flows in fractured porous media

TL;DR

This paper introduces a novel residual-type a posteriori error estimator for adaptive staggered DG methods on polygonal meshes, improving efficiency in simulating Darcy flows in fractured porous media with complex features.

Contribution

It develops and analyzes a new error estimator that supports adaptive mesh refinement and handles general polygonal meshes with hanging nodes for Darcy flow simulations.

Findings

The error estimator is proven reliable and efficient.

Numerical experiments confirm the theoretical results.

The method effectively handles complex fracture networks.

Abstract

Modeling flows in fractured porous media is important in applications. One main challenge in numerical simulation is that the flow is strongly influenced by the fractures, so that the solutions typically contain complex features, which require high computational grid resolutions. Instead of using uniformly fine mesh, a more computationally efficient adaptively refined mesh is desirable. In this paper we design and analyze a novel residual-type a posteriori error estimator for staggered DG methods on general polygonal meshes for Darcy flows in fractured porous media. The method can handle fairly general meshes and hanging nodes can be simply incorporated into the construction of the method, which is highly appreciated for adaptive mesh refinement. The reliability and efficiency of the error estmator are proved. The derivation of the reliability hinges on the stability of the continuous setting in the primal formulation. A conforming counterpart that is continuous within each bulk domain for the discrete bulk pressure is defined to facilitate the derivation of the reliability. Finally, several numerical experiments including multiple non-intersecting fractures are carried out to confirm the proposed theories.