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Intrinsic Finite Element Methods for Fluids on Riemannian Manifolds Compared with Surface FEM

Yongxing Wang

arXiv:2608.29754v1cs.CE

TL;DR

The paper addresses the limited use of fully intrinsic discretisations for incompressible Navier–Stokes equations on Riemannian manifolds. It develops an intrinsic finite element formulation with an energy-stable backward Euler scheme, validates it using Killing-field steady states, and finds advantages over surface FEM while extending to higher-dimensional manifolds.

  • Problem

    Fully intrinsic parameter-coordinate discretisation for incompressible flows on manifolds remains largely unexplored, despite its natural extension beyond embedded surfaces.

  • Method

    The paper derives a metric- and Christoffel-symbol-based intrinsic weak formulation, uses backward Euler time discretisation, and validates it against surface FEM and Killing-field formulations.

  • Results

    The intrinsic formulation is reported to provide efficiency and accuracy advantages over surface FEM for regular geometries while extending naturally to higher-dimensional manifolds.

  • Takeaways & Limitations

    Killing fields provide steady-state benchmarks for assessing long-time convergence, viscous-energy decay, and preservation of the flow’s geometric structure.

  • Takeaways & Limitations

    Surface FEM comparisons are constrained by mesh quality and surface-normal approximation, and the method may be more computationally expensive and excessively dissipative.

Abstract

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We present an intrinsic finite element formulation for the incompressible Navier--Stokes equations on Riemannian manifolds. We derive the corresponding weak formulation and prove that the backward Euler discretisation is energy stable. The proposed framework is validated on several representative manifolds, with particular attention paid to the long-time behaviour of the flow and its convergence to steady-state solutions represented by Killing vector fields. Comprehensive comparisons are performed with the surface finite element method and a corresponding eigenvalue formulation for Killing vector fields. The numerical results demonstrate that the intrinsic formulation provides an accurate, computationally efficient, and geometrically transparent alternative to embedded surface finite element formulations, while naturally extending to higher-dimensional Riemannian manifolds.

1 Introduction

The paper develops an intrinsic finite element framework for incompressible Navier–Stokes flows on Riemannian manifolds, addressing the limited use of parameter-coordinate discretisations. It derives the formulation, proves energy stability, and compares it with surface FEM using Killing fields as long-time benchmarks.

  • Flows on manifolds replace Euclidean differential operators with Riemannian counterparts, including covariant divergence and intrinsic strain-rate tensors.
  • Killing vector fields have zero viscous dissipation and provide benchmarks for preserving the viscous operator’s nullspace and long-time flow structure.
  • Embedded surface FEM dominates recent numerical work, while fully intrinsic parameter-coordinate discretisation remains largely unexplored for incompressible manifold flows.
  • The paper derives an intrinsic weak formulation, proves backward Euler energy stability, and compares intrinsic FEM with surface FEM and an eigenvalue formulation.
  • Killing fields are used as exact steady states, and the formulation is demonstrated on a three-dimensional manifold with the Schwarzschild spatial metric.

2 Navier–Stokes equations on Riemannian manifolds

The manifold Navier–Stokes equations are obtained by replacing Euclidean variables and differential operators with their covariant, metric-dependent counterparts. The resulting formulation involves tangent velocities, covariant derivatives, intrinsic stresses, and curvature-related Laplacians.

  • Velocity is represented by tangent-space components, with covariant derivatives defining convection and incompressibility on the manifold.
  • The Euclidean stress tensor is replaced by a contravariant Boussinesq–Scriven surface stress whose divergence has the required free velocity index.
  • Metric components, inverse metric components, and Christoffel symbols generate additional terms when covariant derivatives expand the equations.
  • The divergence of the viscous strain term yields the Hodge Laplacian, while the formulation also identifies the Bochner Laplacian and Ricci curvature term.
  • The manifold Navier–Stokes system retains the Euclidean structure while replacing partial derivatives with covariant derivatives and adding geometric terms.

3 Finite element weak formulation

The weak formulation is derived intrinsically using the manifold divergence theorem and test functions with metric-compatible index contraction. Surface integration can then be evaluated either on an embedded triangulation or in parameter space.

  • The manifold divergence theorem derives the weak form directly, avoiding explicit manipulation of metric and Christoffel terms in the strong equations.
  • Metric-lowered velocity test functions contract naturally with the strong-form velocity equation, while stress terms are integrated by parts.
  • The weak formulation integrates over the Riemannian surface element and includes boundary terms involving the outward unit normal covector.
  • For two-dimensional surfaces, surface FEM triangulates an embedded surface, whereas the intrinsic approach maps it to a parameter domain and incorporates geometric factors there.

4 Discretisation and energy stability

The paper discretises time with backward Euler and establishes an energy estimate for enclosed flows. The stability proof uses the boundary condition, integration by parts, and standard inequalities to control the discrete energy.

  • Backward Euler is used for temporal discretisation, with time levels separated by the step δt.
  • The energy-stability result assumes an enclosed flow with zero normal velocity at the boundary.
  • The proof integrates the convection term by parts and selects velocity and pressure test functions to obtain the discrete energy relation.
  • The resulting energy estimate is obtained after applying the Cauchy–Schwarz inequality and rearranging the discrete balance.
  • At steady state, vanishing viscous energy and near-constant successive solutions yield a numerically stable long-time estimate.

5 Numerical experiments

Numerical tests on spherical geometries compare intrinsic FEM with surface FEM and eigenvalue formulations. Intrinsic simulations reach Killing-field steady states, while singularities, boundary consistency, and surface discretization affect competing formulations.

  • Experimental setup: Tests cover spherical geometries with and without coordinate singularities, using intrinsic FEM, surface FEM, and eigenvalue formulations.The experiments use Taylor–Hood P2/P1 elements and compare computational frameworks across multiple settings.
  • Hemisphere simulations: The intrinsic formulation drives arbitrary initial flows toward rotational Killing-field steady states on the hemisphere.The initial flow need not be divergence-free in the numerical experiment.
  • Hemisphere simulations: Three initial profiles converge to stable Killing-field steady states without numerical damping or instability during simulations to t = 20.This behavior is consistent with the paper’s energy-stability analysis.
  • Surface FEM comparison: Surface FEM exhibits numerical damping, while mesh refinement achieves a steady state but leaves viscous energy far from zero.The refined result also shows greater kinetic-energy damping than the intrinsic result, by one magnitude.
  • Eigenvalue comparison: Eigenvalue comparisons reveal that intrinsic FEM cannot resolve the second and third Killing modes near parameter-space singularities, whereas surface FEM mixes Killing fields with source–sink modes.The expected three-dimensional zero-eigenvalue nullspace is therefore not correctly recovered by the intrinsic calculation in this test.
  • Method limitations: Surface FEM is significantly slower and reduces eigenvalue and viscous-energy magnitudes only to about 10^-5 with P2 elements and 10^-4 with P1 elements.Its results also depend on accurate surface-normal representation and mesh quality.

5.2 Torus and helicoid

The intrinsic formulation is tested on torus and helicoid geometries with singularity-free parameterisations, where flows converge to stable Killing-field steady states. The simulations also examine mesh resolution, long-time energy behaviour, and extension to higher-dimensional geometries.

  • Motivation: Singularity-free parameterisations make the torus and helicoid suitable tests of the intrinsic formulation’s efficiency and accuracy.The paper identifies both geometries as settings without coordinate singularities.
  • Torus: The torus simulation converges to a stable rotational Killing-field steady state under periodic boundary conditions.The torus uses R = 3 and r = 1, with periodicity in θ.
  • Helicoid: The helicoid simulation converges to a translational Killing-field steady state while viscous energy decays to zero and kinetic energy remains stable.The convergence is shown both in parameter space and after mapping the flow onto the helicoid.
  • Helicoid: The helicoid velocity-field convergence is visualised in parameter space and on the mapped manifold.Figures 19 and 20 compare the parameter-space flow with its R3 representation.
  • Higher-dimensional geometry: The spherical-shell experiment also approaches a stable rotational Killing-field steady state under the reported large-scale parallel discretisation.The simulation uses 60,000 nodes, 300,000 tetrahedra, and approximately 2,000,000 degrees of freedom.

6 Conclusion

The paper develops and validates an intrinsic finite element formulation for incompressible Navier–Stokes flows on Riemannian manifolds. Results show stable convergence toward Killing-field equilibria and advantages over surface FEM when regular parameterisations are available.

  • Method: The formulation is assembled in parameter space, with the metric tensor, inverse metric, and Christoffel symbols carrying the geometric information.The weak formulation is derived, and backward Euler time discretisation is proved energy-stable.
  • Validation: Killing vector fields provide steady-state benchmarks because they form the null space of the strain-rate operator.This makes them suitable for testing long-time numerical behaviour.
  • Comparison: The intrinsic formulation offers efficiency and accuracy advantages over surface FEM for geometries admitting regular parameterisations.It is also geometrically transparent and extends naturally to higher-dimensional manifolds.
  • Numerical results: For regular geometries such as the torus and helicoid, viscous energy decays to zero while kinetic energy remains constant in the long-time regime.The simulations converge stably toward the expected Killing fields.
  • Comparison: Surface FEM requires sufficiently accurate meshes and surface normals, and in these computations it is more expensive and may exhibit excessive numerical dissipation.The comparison identifies practical limitations rather than a universal replacement of one framework by the other.
  • Scope: The intrinsic and embedded surface FEM formulations are complementary: intrinsic FEM suits regular parameterisations and higher-dimensional manifolds, whereas surface FEM suits cases without global parameterisations.Coordinate singularities are identified as a boundary for the intrinsic parameter-space approach.

Declarations

The declarations report no external funding or competing interests and state that supporting data and FreeFEM source code are publicly available.

  • Declarations: The research received no external funding and the author declared no competing interests.
  • Declarations: Supporting data and FreeFEM source code are publicly available through the listed project website.
  • Declarations: Ethics approval, consent to participate, consent for publication, and materials availability were not applicable.

A Geometric preliminaries and covariant derivatives

This section introduces tangent and cotangent bundles, tensor fields, local coordinates, and the Levi–Civita connection used for intrinsic differential geometry. It distinguishes covariant differentiation from ordinary coordinate derivatives and characterizes the connection through metric compatibility and vanishing torsion.

  • Bundles and sections: Tangent and cotangent bundles collect the pointwise spaces T_pM and T*_pM, while smooth sections represent vector fields and one-forms.Vectors in different fibers cannot generally be added because the bundles lack a global vector-space structure.
  • Local coordinates: Local coordinates provide bases for tangent and cotangent spaces and component representations for tensor fields.The section develops these representations before introducing covariant differentiation.
  • Covariant derivatives: Covariant differentiation extends ordinary partial differentiation by accounting for variation of the coordinate basis through Christoffel symbols.These connection coefficients describe how basis vectors change across coordinates.
  • Levi–Civita connection: The Levi–Civita connection is uniquely determined by being torsion-free and compatible with the Riemannian metric.Metric compatibility also gives ∇g = 0 and the product rule for the metric inner product.
  • Levi–Civita connection: The resulting connection coefficients can be computed from metric derivatives together with the torsion-free condition.This provides the coordinate-level construction used by the intrinsic formulation.

B Bochner and Hodge Laplacians

This section derives the relationship between the strain-rate operator and geometric Laplacians on a Riemannian manifold. It emphasizes the distinction between tensor components and iterated directional covariant derivatives.

  • Higher-order derivatives: Second covariant derivatives must distinguish tensor components from successive directional operations such as ∇_j∇_m u.The former represents components of a tensor, whereas the latter is not itself a tensor.
  • Covariant-derivative notation: The covariant derivative of a vector field is a (1,1)-tensor, while ∇_j u and ∇_v u denote coordinate- and direction-specific derived objects.Keeping these notations distinct is necessary for interpreting higher-order derivatives.
  • Hodge Laplacian: Taking the divergence of the strain-rate tensor produces the Hodge Laplacian after index manipulations and use of metric compatibility.Raising indices commutes with covariant differentiation because ∇g = 0.
  • Bochner Laplacian: The first term in the resulting expression is the Bochner Laplacian, while a curvature-related second term generally remains on non-Euclidean manifolds.That second term vanishes in Euclidean space under the divergence-free condition but not on a general manifold.

C Lie derivative

The Lie derivative describes how tensor fields change along flows generated by vector fields on a manifold. Pullbacks compare tensors at different points, while specialized formulas define the derivative for scalars, vectors, forms, and tensors.

  • Flow generated by a vector field: A vector field X generates a flow φ_t(p) that moves an initial point p along the manifold.The initial point specifies the streamline or flow curve.
  • Tensor transport: The Lie derivative of a tensor field T along X is defined using the flow generated by X.The pullback brings tensors back to a common point before comparison.
  • Tensor transport: The pullback operator compares tensors at different points by transporting them back to the initial point.For a 1-form, the pullback acts through the differential of the flow.
  • Specialized formulas: For scalars and vector fields, the Lie derivative is respectively X(f) and the commutator [X, Y].Coordinate expressions are also given for these cases.
  • Specialized formulas: The framework provides coordinate-form expressions for Lie derivatives of 1-forms, (1,1)-tensors, and (2,0)- and (0,2)-tensors.These formulas extend the construction beyond scalar and vector fields.

D Killing vector fields

Killing vector fields preserve the metric under their generated flow, forcing the symmetric part of the covariant velocity derivative to vanish. Consequently, they are divergence-free steady solutions of the incompressible Navier–Stokes equations.

  • Definition and geometry: A vector field is Killing when the Lie derivative of the metric along it vanishes, meaning the metric is unchanged by its flow.This is equivalent to the coordinate Killing condition.
  • Strain and rotation: The Killing condition makes the symmetric part of the covariant derivative vanish, leaving only the antisymmetric rotational part.This decomposition identifies the strain-free character of Killing fields.
  • Strain and rotation: For any Killing field u, the strain-rate tensor satisfies ϵ(u) = 0.Thus the viscous strain contribution vanishes for the field.
  • Incompressibility: Killing fields satisfy the divergence-free condition because contracting the Killing equation with the metric gives 2∇_i u^i = 0.The divergence-free property is established directly from the antisymmetry relation.
  • Steady solutions: Every Killing field provides a steady solution of the incompressible Navier–Stokes equations.The conclusion combines its vanishing strain rate with its divergence-free property.
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