A Simple Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces
Provides an easy-to-implement method for engineers to assess crack orientation effects in Darcy flow, though it is an incremental improvement over existing embedded interface methods.
The paper develops a finite element method for simulating high-permeability embedded layers (cracks) in a lower-permeability matrix, using mesh refinement near cracks to recover optimal convergence. Numerical examples confirm the theoretical error estimates.
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.