A Stabilized Finite Element Method for the Darcy Problem on Surfaces
This work provides a theoretical foundation for solving Darcy flow on surfaces, which is important for applications in geophysics and biology, but the method is an incremental extension of existing formulations.
The authors developed a stabilized finite element method for the Darcy problem on surfaces, using the Masud-Hughes formulation with weak enforcement of the tangential velocity condition. They proved optimal order a priori estimates accounting for geometry and solution approximation.
We consider a stabilized finite element method for the Darcy problem on a surface based on the Masud-Hughes formulation. A special feature of the method is that the tangential condition of the velocity field is weakly enforced through the bilinear form and that standard parametric continuous polynomial spaces on triangulations can be used. We prove optimal order a priori estimates that take the approximation of the geometry and the solution into account.