A cut finite element method for a model of pressure in fractured media

Abstract

We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: (1) Stability of the formulation in the full range of parameter choices; and (2) Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.

Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping

“…For the CIP stabilization this is always true and even for the P2 elements the stability region is very thin. As suggested in [17,38] higher order derivatives jump stabilization terms might fix this problem, but it introduces more parameters. This has not been considered here.…”

“…In this direction, it would be interesting to evaluate the stability of the CIP adding a additional penalty term on the jump of higher order derivatives as suggested in [17,13,3]. Moreover, it could be interesting to see the stability of Cubature elements using higher degree polynomials.…”

“…On the other side, the stencil of the scheme increases as the jump of a degree of freedom interacts with cells which are not next to the degree of freedom itself (up to 2 cells distance). Note that for higher order approximations [17,38] suggest the use of jumps in higher derivatives to improve the stability of the method. However, here we consider the jump in the first derivatives in order to be able to apply the stability analysis and to study the influence of δ on the stability of the method.…”

“…Note that for higher than second order it may be relevant to consider additional penalty terms based on higher derivatives (see e.g. [17,13,3]). We did not do this in this work.…”

Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems

“…This formulation underpins hybrid DG methods [2], where u h is replaced by an independent "hybrid" variable which is particularly useful for model coupling purposes. [6,5] The derivation from an augmented Lagrangian method follows along the same lines as the previous formulation, and we can write the Newton iteration scheme as that of finding ūh ∈ DP such that…”