A Stabilized Cut Finite Element Method for the Darcy Problem on Surfaces
This work provides a stable and high-order numerical method for solving Darcy flow on surfaces, which is important for applications in porous media and biological membranes, but the approach is incremental as it extends existing stabilization techniques to cut finite elements.
The authors developed a stabilized cut finite element method for the Darcy problem on surfaces, achieving high-order accuracy and robust conditioning regardless of surface positioning within the mesh. Numerical experiments confirm optimal convergence rates for the normal gradient stabilized variant.
We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three dimensional mesh as trial and test functions. Since we consider a partial differential equation on a surface, the resulting discrete weak problem might be severely ill conditioned. We propose a full gradient and a normal gradient based stabilization computed on the background mesh to render the proposed formulation stable and well conditioned irrespective of the surface positioning within the mesh. Our formulation extends and simplifies the Masud-Hughes stabilized primal mixed formulation of the Darcy surface problem proposed in [28] on fitted triangulated surfaces. The tangential condition on the velocity and the pressure gradient is enforced only weakly, avoiding the need for any tangential projection. The presented numerical analysis accounts for different polynomial orders for the velocity, pressure, and geometry approximation which are corroborated by numerical experiments. In particular, we demonstrate both theoretically and through numerical results that the normal gradient stabilized variant results in a high order scheme.