A cut finite element method for the Darcy problem

Abstract

We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is based on the Nitsche formulation studied in [12] and can be considered as a first attempt at extension in the unfitted case. The key feature is to add two ghost penalty operators to stabilize both the velocity and pressure fields. We rigorously prove our stabilized formulation to be well-posed and derive a priori error estimates for the velocity and pressure fields. We show that an upper bound for the condition number of the stiffness matrix holds as well. Numerical examples corroborating the theory are included.

A divergence preserving cut finite element method for Darcy flow

“…We present two cut finite element schemes, both employing ghost penalty stabilization. The first method is based on standard ghost penalty terms both for the pressure and the velocity [7,20] but can use a macro-element partition of the active mesh so stabilization is active only where necessary. The second method contains new mixed stabilization terms related to the b-bilinear form, replacing the standard stabilization for the pressure; a macro-element partition can also be utilized here.…”

“…Consider now Γ := ∂Ω, T h is unfitted with respect to Γ, and we seek velocity u u u and pressure p satisfying the following problem from [20] ηu u u + ∇p = f f f in Ω,…”

“…In this example ∂Ω is a circle with radius 0.45. This is the example of a cut circle featured in [20]. We set ∂Ω = ∂Ω u u u .…”

“…A cut finite element discretization of the Darcy problem based on Raviart-Thomas spaces and the standard ghost penalty stabilization for both velocity and pressure, in a setting where the mesh is not fitted to the external boundary, has recently been proposed and analyzed in [20]. Although property 1) and 2) are satisfied by that numerical scheme and thus ill-conditioned linear systems are avoided, numerical experiments clearly show that the divergence-free property of the Raviart-Thomas element is lost and spurious errors are present in the divergence approximation due to the stabilization.…”