A Simple Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

TL;DR

This paper introduces a simple finite element method for simulating flow in embedded cracks within a matrix, using mesh refinement near cracks to achieve optimal convergence, suitable for rapid assessment and optimization.

Contribution

The paper presents a straightforward FEM approach for embedded crack flow that employs mesh refinement based on a priori error estimates to ensure optimal convergence.

Findings

Mesh refinement near cracks improves convergence

Numerical results confirm theoretical error estimates

Method is easy to implement and useful for optimization

Abstract

In this paper we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop.