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

Abstract: This paper presents a pressure-robust and element-wise divergence-free nonconforming finite element method for the Stokes problem on curved domains. The discrete element is constructed by mapping the Fortin-Soulie element from a reference triangle using an isoparametric mapping for the geometry and a Piola transform for the function space. The inf-sup condition and the error estimate with optimal convergence rate are proved. Pressure-robustness is obtained by replacing the discrete velocity test functions with the first-order Raviart-Thomas functions. Numerical examples are provided to validate the theoretical results.

• The paper introduces a nonconforming Fortin–Soulie FEM that achieves pressure-robustness and divergence-free velocity on curved 2D domains.
• It employs isoparametric mapping and Piola transforms to maintain optimal error estimates and discrete inf-sup stability.
• The method demonstrates third-order L2 accuracy for velocity and second-order H1 accuracy, verified by detailed numerical experiments.

Pressure-Robust Nonconforming Fortin–Soulie FEM for the Stokes Equation on Curved Domains

Introduction and Motivation

The accurate discretization of incompressible flow problems over curved domains remains an outstanding challenge in computational fluid dynamics and numerical PDE theory. For the Stokes equations, element-wise divergence-free finite element methods (FEMs) provide pressure-robustness and optimal approximation properties when the domain is polygonal. However, their extension to smooth or curved geometries is hindered by loss of conformity, reduced approximation order, and complications in enforcing divergence constraints.

This paper introduces a comprehensive framework for constructing pressure-robust, element-wise divergence-free, nonconforming Fortin–Soulie finite element methods for the Stokes problem on two-dimensional curved domains (2604.12769). The approach employs a combination of isoparametric geometric mappings, Piola transformations on function spaces, and pressure-robust reconstruction operators to establish discrete inf-sup stability, optimal error estimates, and pressure-velocity decoupling in the presence of geometric curvature.

Finite Element Construction on Curved Domains

The principal development is the isoparametric mapping of the classical Fortin–Soulie nonconforming element from reference triangles to curved physical subdomains, accompanied by the Piola mapping for vector-valued function transformation. Specifically:

• Geometric Mapping: Each element is mapped via an isoparametric quadratic transformation $F_T$ from the reference triangle $\hat{T}$ to its curved physical counterpart, ensuring $\mathcal{O}(h^2)$ geometric approximation and shape regularity.
• Function Space Mapping: The finite element basis is constructed by mapping the reference element’s polynomial basis via the contravariant Piola transform $A_T$, ensuring that element-wise divergence is preserved under change of coordinates.
• Nonconforming Setting: The finite element functions are only continuous (i) at edge endpoints and midpoints (quadratic nodal points) for the mapped quadratic Lagrange subspace; (ii) at Gauss–Legendre quadrature points for nonconforming bubble functions, leading to the lack of interelement continuity except at selected locations. Nevertheless, continuity at enough points allows controlling consistency errors.

The global velocity space $V_h$ is then set as the direct sum of the mapped conforming quadratic subspace and a mapped nonconforming bubble subspace, paralleling the decomposition for the original Fortin–Soulie triangle but with transformations that accommodate curved geometry.

Discrete Variational Problem and Stability

The discrete scheme employs the pair $(V_h, Q_h)$ where $Q_h$ is the mapped piecewise linear (isoparametric) pressure space. The variational formulation is:

• Velocity Equation: Seek $(u_h, p_h) \in V_h \times Q_h$ such that

$$\int_{\Omega_h} \nu \nabla_h u_h : \nabla_h v_h - \int_{\Omega_h} p_h \operatorname{div}_h v_h = \int_{\Omega_h} f_h \cdot v_h, \quad \forall v_h \in V_h$$

• Divergence Constraint:

$$\int_{\Omega_h} q_h \operatorname{div}_h u_h = 0, \quad \forall q_h \in Q_h$$

A notable theoretical result is the proof that the divergence constraint is satisfied elementwise, even on curved geometries, thus maintaining pressure-robustness at the discrete level—a property typically lost in isoparametric or standard nonconforming extensions.

Inf-sup stability is established via a two-step process controlling both the mean-value (piecewise constant) and oscillatory components of the discrete pressure. The approach utilizes special nonconforming bubble functions and their mapped counterparts to enforce the divergence-free condition, and it adapts arguments from the Scott–Vogelius theory for reference elements to the curved setting via mapped basis and quadrature.

Error Analysis and Pressure-Robustness

The error analysis demonstrates that the method achieves optimal rates of convergence for velocity and pressure in appropriate Sobolev norms, under mild regularity assumptions on the domain and data. Specifically, for exact solutions $\hat{T}$0, the velocity error in the $\hat{T}$1-seminorm satisfies:

$\hat{T}$2

If the standard right-hand side is used ($\hat{T}$3), the error constant is inversely proportional to viscosity, manifesting a coupling of velocity convergence to pressure errors. To mitigate this, the authors introduce a pressure-robust reconstruction: the test functions on the right side are replaced by their projection to the lowest-order parametric Raviart–Thomas space using a carefully constructed operator $\hat{T}$4. This modified scheme yields a velocity error:

$\hat{T}$5

This is independent of the pressure or viscosity scale, confirming strong pressure-robustness, which is corroborated by numerical experiments.

Implementation Aspects and Numerical Results

The paper details the practical construction of isoparametric curved meshes compatible with the mapping framework, as well as the realization of the velocity reconstruction operator on curved elements using parametric Raviart–Thomas spaces. Several technical lemmas are provided for extensions of norm equivalence, quadrature, and interpolation bounds under nonlinear mappings.

Numerical experiments validate the theoretical predictions:

• No-flow Test: For irrotational forcing (pure pressure-driven), the modified scheme returns a numerically vanishing velocity, consistent with theory, while the standard method produces spurious velocities.
• High-pressure-gradient Case: As viscosity decreases, the standard approach’s velocity errors are inflated proportionally to $\hat{T}$6, while the modified method's errors are unchanged across several orders of magnitude, evidencing pressure-robustness.
• Convergence Rates: Both schemes achieve third-order accuracy in $\hat{T}$7 and second-order in $\hat{T}$8 for velocity and pressure, matching estimates.

Discussion and Theoretical Implications

This work extends divergence-free, pressure-robust finite element methods to curved 2D domains within the nonconforming family, addressing the longstanding issue of accuracy loss and loss of divergence-free property for isoparametric methods. Unlike previous isoparametric methods that sacrifice divergence control or succumb to geometric inconsistency, the proposed framework maintains both divergence-free velocity and pressure-robustness through a rigorous combination of geometric mapping and function space transformations.

A key theoretical contribution is the demonstration that reconstruction-based pressure-robust FEMs, previously known for polygonal domains [linke2016robust, linke2014on], can be systematically extended to curved geometries through isoparametric-piola mapping and careful quadrature management. This addresses the critical issue that direct generalization to 3D remains nontrivial due to the lack of necessary quadrature rules for bubbles on curved tetrahedra, leaving robust 3D extension as an open problem [li2025divergence].

Future Directions

Potential avenues for future research include:

• Higher-Order and 3D Generalization: Extending the framework to higher-order elements and three-dimensional smooth domains will likely require new theoretical tools, such as the adoption of parametric Brezzi–Douglas–Marini elements and DG stabilization.
• Extension to Navier–Stokes: Adapting the scheme for nonlinear (Navier–Stokes) flows involves tackling the convective nonlinearity in the presence of arbitrary domain curvature.
• Robust A-Priori and A-Posteriori Error Estimation: Refinement of error analysis for adaptive schemes and error estimators under the mapping framework.
• Complex Geometries and Boundary Conditions: Addressing nontrivial boundary layers and slip/non-slip conditions on geometrically complex surfaces.

Conclusion

This paper provides a mathematically rigorous and computationally implementable framework for pressure-robust, divergence-free, nonconforming finite element methods on curved domains for the Stokes equations (2604.12769). By leveraging isoparametric geometric mapping, discrete Piola transformations, and pressure-robust reconstructions, the authors establish optimal convergence and stability results, verified by numerical tests. The methodology offers a path forward to robust, high-accuracy incompressible flow simulations in domains of practical geometric complexity and poses significant theoretical and computational questions for future development, particularly in the three-dimensional setting.