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A Pressure-Free Virtual Element Method for the Surface Stokes Problem

Jun Hu, Shengyang Xu, Hao Zhou

arXiv:2609.39996v2math.NAcs.NA

TL;DR

Surface Stokes methods on polygonal meshes must handle tangential continuity, incompressibility, and pressure robustness, including harmonic components on surfaces of positive genus. The paper extends a planar VEM Stokes complex with an edge correction and divergence-preserving load reconstruction, and also constructs an equivalent macroelement formulation. Both formulations are stable and achieve optimal velocity convergence rates.

  • Problem

    On piecewise-planar meshes, differing face tangent planes complicate imposing tangentiality and interelement continuity together, while positive-genus surfaces require harmonic components beyond a global stream function.

    Source: §1 · pp. 1–2

  • Method

    The paper extends the planar VEM Stokes complex to polygonal surfaces, adding an edge correction and a divergence-preserving load reconstruction, and constructs an equivalent macroelement formulation.

    Source: §1 · pp. 1–2, §4.1 · p. 13, §5.2 · p. 19

  • Results

    Both formulations are stable and achieve optimal velocity convergence: first order in broken H1 and second order in L2.

    Source: §1 · p. 2, §8 · p. 28

  • Takeaways & Limitations

    The edge correction provides weak continuity while preserving the complex without adding independent degrees of freedom.

    Source: §8 · pp. 27–28

    Without the shared edge constraint, the consistency term has only an O(h) bound, so the second-order weak estimate needed for L2 error analysis is not guaranteed.

    Source: §6.1 · p. 22

Abstract

from arXiv · show

We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method extends the planar VEM Stokes complex to discrete surfaces, yielding a nonconforming complex with an exactly tangential, $H(\operatorname{div})$-conforming velocity space, commuting interpolation, and the correct discrete cohomology. To obtain the weak consistency needed for optimal convergence, we introduce an edge correction that preserves the complex without adding degrees of freedom; the resulting construction reduces to the original complex in the planar setting. The virtual formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct explicit macroelement representatives whose energy and load induce an alternative formulation on the VEM space, algebraically equivalent to a direct macroelement Galerkin method. The original virtual formulation and the macroelement method are stable and attain optimal first-order convergence in the broken $H^1$ norm and second-order convergence in the $L^2$ norm for the velocity. Numerical experiments illustrate these convergence rates, the role of the edge correction, and pressure robustness.

1 Introduction

Surface Stokes discretization must reconcile tangentiality with continuity on piecewise-planar meshes, while pressure-free methods must represent harmonic components on positive-genus surfaces. The paper extends the planar VEM Stokes complex and adds an edge correction to obtain a pressure-free, penalty-free method with robust loads and optimal velocity convergence.

1 Introduction

  • The resulting velocity space is exactly tangential and H(div)-conforming, with commuting interpolation and the surface’s cohomology dimension preserved.These topological components allow pressure elimination on closed surfaces of arbitrary genus.Source: §1 · p. 2
  • The edge correction enforces shared tangential edge averages without extra degrees of freedom, supplying weak continuity while preserving the complex and reducing to the planar complex.Vertex coupling alone is insufficient for the weak continuity needed for optimal L2 approximation.Source: §1 · p. 2
  • A local divergence-preserving reconstruction makes the load computable for general L2 forces and pressure-robust: adding a gradient leaves the discrete velocity unchanged.The discrete problem is posed on the full divergence-free space and needs neither normal-component nor edge-jump penalties.Source: §1 · p. 2
  • Explicit representatives define an alternative virtual-space formulation algebraically equivalent to direct macroelement Galerkin, including divergence-free spaces and topological components.The representatives preserve boundary data and differential structure.Source: §1 · p. 2
  • Both formulations are stable and converge at first order in broken H1 and second order in L2 for velocity.For pure gradient forces, the broken H1 error decays at second order; experiments test this estimate and the edge correction.Source: §1 · p. 2

2 Preliminaries

The preliminaries establish the surface geometry, differential operators, and mesh assumptions used to define the discrete spaces. They then describe transforms between the curved surface and polygonal mesh and introduce the local spaces forming the VEM Stokes complex.

2 Preliminaries

  • The analysis assumes a C4 closed surface and an admissible polygonal mesh whose closest-point map is bijective and whose elements are uniformly star-shaped.The mesh assumptions also provide uniform geometric bounds on element sizes and local face counts.Source: §2 · p. 3, §2.1 · p. 3

2.1 Surface geometry and differential operators

  • Surface gradient, divergence, and strain are defined through closest-point extension and tangential projection; discrete differential operators act facewise.The mesh consists of planar polygonal faces approximating the closed surface.Source: §2.1 · pp. 3–4

2.2 Surface Piola and vertex transforms

  • The surface Piola transform preserves normal traces, while vertex transforms couple velocities across faces with differing tangent planes.The vertex-transform comparison and tangential-jump estimates support assembly across the polygonal surface.Source: §2.2 · pp. 4–5

2.3 The local virtual element spaces

  • The local velocity and scalar spaces have unisolvent boundary degrees of freedom and form an exact Stokes complex on each simply connected polygon.The local spaces have no cell degrees of freedom; divergence-free local velocities are curls of scalar functions in the scalar space.Source: §2.3 · p. 6

3 A nonconforming Stokes complex on polygonal surfaces

The paper constructs a nonconforming Stokes complex on polygonal surfaces that preserves the expected topology and provides commuting interpolation. An edge correction supplies weak continuity while retaining the planar construction as a special case.

3.1 Global spaces and the discrete complex

  • The assembled spaces preserve H1–H(div)–L2 conformity, with higher-order jump estimates providing approximate continuity in the remaining trace components.The global construction assembles local spaces using vertex transformations, normal-flux continuity, and edge integral constraints.Source: §3.1 · pp. 7–8
  • The edge correction enforces zero-mean tangential jumps through shared edge means without adding independent degrees of freedom.It recovers the cancellation needed for second-order weak consistency and reduces to the original planar construction on planar meshes.Source: §3.1 · p. 9
  • For connected, closed, orientable surfaces, the discrete complex is exact except possibly at the velocity space, where cohomology has dimension 2 − χ(Γh).The dimension is given by the mesh Euler characteristic χ(Γh) = NV − NE + NF.Source: §3.1 · p. 9

3.2 Commuting interpolation and approximation estimates

  • The velocity, scalar, and pressure interpolants commute with the discrete differential operators.In particular, the interpolated velocity's discrete divergence equals the interpolated continuous divergence.Source: §3.2 · p. 11

4 The virtual element method

The virtual element method solves the discrete Stokes problem on the full divergence-free space and constructs computable forms and loads. A local divergence-preserving reconstruction yields a pressure-robust load, and stability gives a unique solution for sufficiently small mesh size.

4.1 The discrete formulation

  • The discrete formulation is pressure-free: it solves on the full divergence-free space, with a computable bilinear form requiring neither normal-component nor edge-jump penalties.A curl-space and finite-dimensional complement decomposition provides an algebraic formulation of the solve.Source: §4 · p. 12, §4.1 · p. 13

4.2 Pressure-robust load discretization

  • A local reconstruction produces a globally divergence-free field, making the load computable from L2 data and ensuring it annihilates gradient forces.The reconstruction uses a piecewise quadratic H(div) space with the full normal trace prescribed.Source: §4.2 · p. 14
  • The reconstructed field belongs to H(divΓh; Γh) and has zero surface divergence.Normal-trace conformity across surface edges and continuity across interior fan edges yield the global conformity.Source: §4.2 · p. 15
  • The paper also constructs piecewise-polynomial representatives whose energy and load give an alternative pressure-robust formulation.This alternative formulation is algebraically equivalent to the direct macroelement Galerkin method.Source: §4.2 · p. 15

4.3 Stability

  • For sufficiently small h, the discrete problem has a unique solution, with its broken H1 norm bounded by the L2 norm of the force.The bound follows from coercivity and boundedness of the load.Source: §4.3 · p. 16

5 An explicit computational representative

The section constructs explicit macroelement representatives for the implicit virtual spaces, then uses them to define a computable alternative formulation. Trace-preserving isomorphisms give an exact correspondence with a direct macroelement Galerkin method.

5.2 A macro-induced variational realization

  • The alternative formulation evaluates energy and load through explicit piecewise-quadratic velocities on HCT subtriangles, without projection or stabilization.It is an alternative to the original virtual formulation and uses a different energy and load.Source: §5 · p. 16, §5.2 · p. 18

5.1 A macroelement representative of the complex

  • The scalar representative uses HCT functions matching cubic value and quadratic gradient boundary traces, with a unique interior extension.The Hessian seminorm is a norm on the constrained interior space.Source: §5.1 · p. 17
  • The representative spaces are isomorphic to the virtual spaces, preserve full boundary traces, and form a commuting diagram of exact complexes.The global macro spaces retain the virtual spaces’ degrees of freedom.Source: §5.1 · pp. 17–18
  • The representative preserves stability and approximation, including an interpolation estimate for H^2 tangential fields that remain in the divergence-free macro space.Under the stated assumptions, the combined L2 and mesh-scaled broken H1 interpolation error is bounded by h^2 times the H^2 norm.Source: §5.1 · p. 18

5.2 A macro-induced variational realization

  • The macro-induced virtual and direct macroelement problems are well posed and have the same stiffness matrix and load vector in paired bases.Their solutions are identified by the representative-space isomorphism.Source: §5.2 · p. 19

6 Error estimates

The error analysis bounds load, VEM-form, and geometric consistency errors under the stated mesh, load, and regularity assumptions. It establishes optimal error estimates for the virtual and macroelement formulations, while identifying why the edge correction is needed for second-order weak consistency.

6.1 Consistency errors

  • The discrete-load and VEM-form consistency analysis separates general test-function bounds from sharper bounds for interpolated tests.The supplied estimates include O(h) for general tests and O(h^2) for interpolated tests in the cited consistency terms.Source: §6.1 · p. 21
  • Geometric consistency is controlled by bounds for the broken bilinear form and surface mass pairing under the mesh assumptions.The cited bounds scale as h in the broken H1 norms and h^2 in the L2 norms.Source: §6.1 · p. 21
  • The edge correction removes an extra consistency contribution that otherwise admits only an O(h) bound, so second-order weak consistency is not guaranteed for uncorrected velocities.The shared edge constraint eliminates this contribution in the consistency argument.Source: §6.1 · p. 22

6.2 Broken H1 and L2 error estimates

  • For a gradient force, the continuous velocity is zero and the discrete velocity satisfies an h^2 broken H1 bound.The bound is proportional to the L2 norm of the gradient force.Source: §6.2 · p. 23
  • Both macroelement formulations also satisfy the optimal velocity error estimates under the mesh and load approximation assumptions.The section establishes estimates in the broken H1 and L2 norms.Source: §6 · p. 19, §6.2 · p. 24

7 Numerical experiments

Numerical experiments on surfaces of different genera confirm the methods’ convergence rates and test pressure robustness and the edge correction. The edge-corrected velocity space improves L2 convergence over the uncorrected space on the torus.

Necessity of shared edge means

  • On the torus, Σh has second-order L2 convergence while eΣh has first-order L2 convergence; both spaces have approximately first-order energy convergence.The comparison uses the same triangular meshes and velocity degrees of freedom.Source: §7 · p. 25, Table 1 · p. 25

Genus-two surface

  • The genus-two example has a velocity with nonzero harmonic component, since its circulation around a surface loop is π.This rules out a globally single-valued stream function for that velocity.Source: §7 · p. 25

Convergence

  • Both methods show approximately first-order energy-error and second-order L2-error convergence on the tested simply connected, torus, genus-two, and genus-five surfaces.Tables 2 and 3 compare the methods on the same meshes.Source: §7 · p. 26, Table 2 · p. 27, Table 3 · p. 27

Pressure robustness

  • For pure gradient forces with exact velocity zero, the VEM energy error decays at second order on both the torus and genus-two surface.The experiment uses f = ∇γq with q = ez/2.Source: §7 · p. 27, Table 4 · p. 27

8 Conclusions and discussion

The paper extends the planar Stokes complex to polygonal surfaces and adds an edge correction that supports optimal convergence without extra degrees of freedom. It also identifies a computability challenge for higher-order VEMs on curved geometries and a possible route for future work.

Contributions

  • The extended complex captures the topological components of the full divergence-free space, enabling pressure elimination on closed surfaces of arbitrary genus.The construction extends the planar Stokes complex to polygonal surfaces.Source: §8 · p. 27
  • Shared tangential edge means provide second-order weak consistency while preserving the complex and adding no independent degrees of freedom.The virtual formulation and the algebraically equivalent direct macroelement method are stable and achieve optimal convergence rates.Source: §8 · p. 28

Computability and future work

  • On curved geometries, variable metric and tangential-projector weights can make local integrals unavailable from standard polynomial moments, challenging higher-order VEM computability.The usual VEM projection argument may therefore be insufficient to assemble local forms.Source: §8 · p. 28
  • Explicit representatives determined by virtual degrees of freedom can be evaluated and differentiated at quadrature points, making geometry-weighted energy and load integrals computable by numerical quadrature.Future work will analyze how representatives, geometric approximation, and quadrature jointly affect stability and accuracy.Source: §8 · p. 28

A Proof of Lemma 6.4

The proof establishes estimates for the geometric and discrete strain terms by applying prior bounds, differentiating the mapped fields, and controlling the resulting remainders. Density extends the calculation to the stated spaces.

Proof of Lemma 6.4

  • The proof obtains estimates (6.6a) and (6.6b) by applying prior results to the Piola pullback and using equation (2.9).Estimates (6.6c) and (6.6d) follow similarly from a cited lemma and its proof.Source: A Proof of Lemma 6.4 · p. 28
  • Differentiating the cofactor form of the pulled-back field retains the term involving the derivative of the distance function.The passage sets up this differentiated expression for the subsequent bounds.Source: A Proof of Lemma 6.4 · p. 28
  • Tangentiality and polynomial differentiation bound the normal derivative and Taylor coefficients; the cited estimate then yields the required bound.The calculation extends to the stated spaces by density.Source: A Proof of Lemma 6.4 · p. 28
  • Expanding the discrete strains around their extended smooth strains controls mixed terms with (A.1) and bounds the product of defects by order h^2.Together with the area bound and (6.6d), this proves (6.6e).Source: A Proof of Lemma 6.4 · p. 28

B Geometric edge estimate

The geometric edge estimate is proved by decomposing the edge sum into three terms and bounding each by order h^2. The bounds use integration by parts, trace and flux estimates, and cancellation from normal-flux continuity.

Geometric edge estimate

  • The proof decomposes the negative edge sum into J1, J2, and J3 using normal-flux continuity and symmetry of A.This decomposition organizes the geometric edge estimate into separately controlled contributions.Source: B Geometric edge estimate · p. 29
  • Elementwise integration by parts and the cited divergence-pairing estimate bound J1 by order h^2 times the norms of A and z.The estimate also uses (A.1) and Green’s formula on γ.Source: B Geometric edge estimate · p. 29
  • A trace bound permits replacing the face average by the edge value in J2 up to an order-h^2 error, after which a cited flux estimate bounds J2..Source: B Geometric edge estimate · p. 29
  • Common terms cancel in J3 by normal-flux continuity, and tangentiality plus the weak normal estimate bound the remaining term by order h^2.Combining the three bounds proves (B.1).Source: B Geometric edge estimate · p. 29

C Macroelement error estimates

The macroelement error estimates are established for both the macroelement and VEM formulations. The proof combines geometric consistency and interpolation bounds with Strang’s lemma and a duality argument.

C Macroelement error estimates

  • The local extension construction selects an affine polynomial by a second-order Poincaré inequality, prescribes boundary HCT data, and sets interior HCT data to zero.The resulting function has the same Cauchy trace as ϕ − p.Source: C Macroelement error estimates · p. 29
  • The geometric-consistency argument uses interpolation estimates; when z_h = I_Σz, it bounds the defect by h∥z∥₂,γ.Source: C Macroelement error estimates · p. 30
  • For the macroelement formulation, Strang’s lemma and geometric consistency establish the H1 error estimate, while duality yields the L2 estimate.The L2 proof uses the duality decomposition from Theorem 6.9; one term vanishes because the macroelement method uses the form a_h itself.Source: C Macroelement error estimates · p. 30
  • For the VEM formulation, interpolation estimates and (5.5) establish the H1 bound, which combines with the macroelement L2 estimate to prove the L2 bound.Source: C Macroelement error estimates · p. 30
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