Surface Stokes Without Inf-Sup Condition

TL;DR

This paper introduces a novel numerical method for surface Stokes equations on hypersurfaces that avoids the traditional inf-sup condition by reformulating the problem as an elliptic system, ensuring stability and accuracy.

Contribution

The authors develop a lifted parametric finite element method for surface Stokes equations that guarantees well-posedness and optimal error estimates without the inf-sup condition.

Findings

Method achieves quasi-best approximation in a robust H^1-norm.

Optimal L^2 error estimates for velocity and pressure.

Numerical experiments confirm accuracy and practicality.

Abstract

For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis for a numerical method based on lifted parametric FEM. Assuming no geometric error for simplicity, we prove its well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as an optimal $L^2$ error estimate 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. We present numerical experiments for velocity-pressure pairs with equal and disparate polynomial degrees, demonstrating that the proposed method is both accurate and practical.