A stabilized dual-SAV parametric finite element framework for constrained planar geometric flows with mesh regularization

Abstract: Parametric finite element discretizations of constrained geometric flows must simultaneously address high-order geometric stiffness, mesh degeneration, and nonlinear global constraints. This paper develops a stabilized dual-SAV (scalar auxiliary variable) parametric finite element framework for planar closed curves. The proposed formulation introduces separate auxiliary variables for the physical geometric energy and for an artificial mesh regularization energy. The mesh regularization is coupled only to tangential motion by projecting out its normal variation, so that mesh redistribution changes the parametrization without introducing an artificial normal driving force. Based on this dual-energy structure, we construct a semi-implicit frozen-metric scheme with zero-order stabilization. The scheme leads to linear spatial response problems and satisfies discrete dissipation estimates for the modified geometric and mesh SAV energies. Nonlinear global constraints are handled by an algebraic block reduction: after solving a small number of symmetric positive-definite response problems, the remaining nonlinear system involves only the geometric auxiliary variable and the Lagrange multipliers. For $K$ global constraints, this reduced nonlinear system has dimension $K+1$; in particular, simultaneous area and length constraints lead to a three-dimensional nonlinear system, independently of the number of mesh vertices. Numerical experiments for curve shortening, area-preserving curve shortening, curve diffusion, and Helfrich-type flows illustrate the modified-energy dissipation, the enforcement of geometric constraints, and the improvement of mesh quality for both second- and fourth-order examples.