Pressure-Robust Fortin-Soulie Elements of the Stokes Equation on Curved Domains

The Stokes equations describe slow-moving viscous fluids and are hard to solve numerically because pressure can pollute velocity accuracy and mass may not be conserved exactly. Standard methods often produce velocity errors that depend on how well pressure is approximated. This work takes the Fortin-Soulie element, which works well on straight triangles and is already divergence-free inside each element, and adapts it to domains with curved boundaries. An isoparametric map approximates the curved geometry while a Piola transform moves the vector fields so that key properties survive the mapping. To make the method pressure-robust, the test functions for velocity are swapped for first-order Raviart-Thomas functions. The authors prove that the resulting discrete problem remains stable (satisfies the inf-sup condition) and that velocity and pressure errors converge at the expected optimal rates as the mesh is refined. Numerical tests on example problems are said to match the theory.