A Stochastic Mixed Formulation for Darcy-Forchheimer Flow on Smooth Surfaces with Mean-Square Stable Surface Finite Element Approximation

We propose a stochastic extension of the surface Darcy–Forchheimer problem on a smooth closed surface Γ ⊂ R3. The unknowns are a tangential velocity field and a scalar pressure, as in the deterministic mixed model, but the momentum balance is perturbed by an additive tangential Wiener forcing. The divergence constraint is preserved pathwise, and the pressure remains a Lagrange multiplier enforcing mass conservation. After rewriting the problem on the divergence-free manifold by means of a right inverse of the surface divergence, we prove well-posedness of a variational solution by monotone-operator methods in expectation and almost surely. We then introduce a lowest-order Raviart–Thomas / piecewise constant discretization on a polyhedral approximation Γh of the surface and couple it with an implicit Euler–Maruyama time stepping. Under standard geometric approximation assumptions, Lipschitz covariance regularity of the noise, and suitable regularity of the exact solution, we derive a strong mean-square error estimate of order O(h + τ 1/2) for the velocity in the natural mixed norm and of order O(h + τ 1/2) for the pressure in L2. The analysis isolates the nonlinear monotonicity term, the stochastic consistency defect, and the geometric perturbation terms induced by the surface approximation. We conclude with modeling perspectives for uncertainty quantification in flows on biological membranes, fractured interfaces, and porous shells.