Abstract: For a d-dimensional hypersurface without boundary, we reformulate the (stationary) tangent Stokes, transient tangent Stokes, and transient tangent Navier–Stokes equations by replacing the divergence equation with a pressure Poisson equation, thereby circumventing their saddle-point structure. The pressure Poisson equation hinges on a commutator identity between the divergence and the surface diffusion operator, whose lower-order term depends on the Ricci curvature. This results in systems with elliptic structure being non-symmetric indefinite and governed by two Laplacians. They are further semidiscretized by ParamatericFEM or TraceFEM. We prove well-posedness, quasi-best approximation in a robust mesh-dependent norm for any polynomial degree, as well as optimal order-regularity error estimates for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. This is joint work with M. Shakipov.

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