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Error analysis of penalized $H({\rm div})$-conforming finite element methods for surface Stokes problems

Alan Demlow, Orsan Kilicer

arXiv:2609.28729v1math.NAcs.NA

TL;DR

Standard conforming velocity spaces cannot simultaneously enforce tangentiality and H1 conformity on typical polyhedral surface approximations. This paper analyzes penalized H(div)-conforming BDM methods, proving energy-norm error estimates and optimal L2 velocity estimates; the lowest-order L2 proof has a stricter-than-usual mesh-placement condition.

  • Problem

    On typical polyhedral approximations, standard conforming velocity spaces cannot simultaneously enforce tangentiality and H1 conformity, while earlier analyses did not fully address geometric errors.

    Source: §1 · pp. 2–3

  • Method

    The paper analyzes penalized H(div)-conforming BDM finite element methods for surface Stokes equations over arbitrary-degree finite element spaces and surface approximations.

    Source: §1 · pp. 3–4

  • Results

    The paper provides a complete energy-norm error analysis and optimal L2 velocity estimates, and experiments show expected isoparametric convergence rates.

    Source: §1 · p. 3, §5.1 · p. 40

  • Takeaways & Limitations

    The analysis applies to BDM_r spaces and piecewise-P_k surface approximations for r,k≥1.

    Source: §1 · p. 3

    For k=1, the optimal L2 proof requires a stricter-than-usual mesh-node placement condition, though experiments suggest it may not be necessary.

    Source: §1 · p. 3

Abstract

from arXiv · show

In recent years a number of finite element methods have been developed for the surface Stokes equations. Constructing standard conforming surface finite element methods for the velocity-pressure formulation of this problem would require simultaneously enforcing $H^1$ conformity and tangentiality of the velocity field, which is not possible on the generalized polyhedral approximating surfaces on which surface FEM are typically posed. Thus methods for this problem weakly enforce either tangentiality or $H^1$ conformity of the velocity field. In this paper we provide an error analysis for an $H({\rm div})$-conforming method based on classical Brezzi-Douglas-Marini elements. In this method continuity of the velocity field is weakly enforced using an interior penalty-type formulation. Advantages of this formulation include exact enforcement of the incompressibility constraint and its construction based on a standard mixed finite element pair. Error analysis of this method in previous work was only valid for lowest-order elements and did not account for ``geometric errors'', i.e., variational crimes due to approximation of the continuous surface on which the PDE is posed by an approximating surface on which the finite element method is posed. Here we provide a full error analysis in a natural discontinuous Galerkin energy norm, with geometric errors accounted for. Optimal estimates for the $L_2$ velocity error are also presented. In the case of lowest-order (polyhedral) surface approximations, our proof of optimal $L_2$ estimates requires a stricter than usual condition on the placement of mesh nodes relative to the surface. Numerical experiments are used to illustrate our results.

1. Introduction

The paper analyzes penalized H(div)-conforming BDM methods for surface Stokes equations, addressing the challenge that standard conforming velocity spaces cannot simultaneously enforce tangentiality and H1 conformity on polyhedral approximations. It provides energy- and L2-error analysis that accounts for surface-approximation errors, with a stricter mesh-placement condition in the lowest-order L2 proof.

1. Introduction

  • Standard conforming velocity spaces cannot enforce both tangentiality and componentwise H1 conformity on the Lipschitz polyhedral surfaces used in typical surface FEM.The paper describes methods that instead weakly enforce either H1 conformity or tangentiality.Source: §1 · p. 2
  • The method uses standard BDM mixed spaces and an interior-penalty formulation to account for the lack of tangential continuity, while exactly enforcing incompressibility.Earlier analyses covered only lowest-order elements or did not rigorously address geometric errors, inf-sup stability, and pressure error estimates.Source: §1 · pp. 2–3
  • The analysis covers BDM_r spaces and piecewise-P_k surface approximations for r,k≥1, with a complete energy-norm error analysis.The DG energy estimate separates Galerkin error from geometric error; it is optimal for isoparametric k=r elements, and optimal L2 velocity estimates are also provided.Source: §1 · p. 3
  • For lowest-order approximations, the optimal L2 proof requires edge-endpoint signed-distance differences of order h^3, stricter than the usual O(h^2) surface-distance condition.The condition holds, for example, when vertices lie on the surface or have a small or consistent O(h^3) bias; experiments suggest it may not be necessary for optimal convergence.Source: §1 · p. 3
  • The analysis compares H1-type forms of Piola-transformed fields and includes geometric errors from penalty terms, discrete Korn-type inequalities, and an inf-sup stability proof.The authors identify these comparisons and stability results as additional challenges in the penalized method’s analysis.Source: §1 · pp. 3–4

2. Preliminaries

The preliminaries set out the surface geometry, discrete surface approximations, transformations, norms, finite element spaces, and bilinear forms used in the analysis. They emphasize Piola transforms and the additional challenges introduced by DG jump and average terms.

2. Preliminaries

  • The preliminaries establish surface-geometry and transformation tools, noting that DG jump and average terms add analysis challenges for the method.The exposition largely follows prior work on H(div)-conforming surface Stokes methods.Source: §2 · p. 4

2.1. Geometric preliminaries

  • The continuous surface is assumed smooth, closed, orientable, and embedded in R3; tangential gradients, divergence, and rate of strain are defined intrinsically.The signed-distance function and closest-point projection provide geometric descriptions of the surface and its neighborhood.Source: §2.1 · pp. 4–5

2.2. Surface approximations, meshes, and reference mappings

  • The surface mesh is built from a shape-regular triangulation and a polynomial map Ψ, with quasi-uniform mesh size and geometric approximation assumptions.The framework includes polyhedral and higher-degree surface approximations, along with lifts and extensions between discrete and continuous surfaces.Source: §2.2 · pp. 5–6

2.3. Piola transforms

  • Piola transforms map vector fields between surfaces and play a central role in relating mapped and transformed functions in the analysis.The stated transform properties include bounds for transforms and their inverses on the mesh elements.Source: §2.3 · pp. 8–9

2.4. Norms and function spaces

  • The framework defines discrete norms and function spaces, including edge jumps and averages, and uses a penalty parameter ρ bounded below independently of h.The norms are used to compare discrete velocity fields across the approximating and exact surfaces.Source: §2.4 · pp. 9–10

2.5. Finite element spaces

  • The velocity finite element space is constructed from BDM_r elements using H(div)-conforming Piola mappings on the reference triangle.The supplied definition places the discrete fields in H(div, Γ̄_h) element by element.Source: §2.5 · p. 10

2.6. Bilinear forms

  • The bilinear-form setup uses jump terms whose conormal components disappear for the H(div)-conforming fields considered, and integration by parts gives consistency.The chosen definition omits the conormal jump, simplifying the analysis while retaining consistency.Source: §2.6 · pp. 11–12

3. Norm equivalences, Korn inequalities, and geometric errors

This section develops norm-equivalence and Korn tools for relating functions on the discrete and continuous surfaces, then uses them to define the BDM interpolant and control geometric errors. The geometric-error bounds include three summed edge contributions, with the final edge estimate relying on a condition on mesh-node placement.

3.1. Equivalence of jumps

  • Jump and L2 portions of the DG norm are equivalent across the discrete and continuous surfaces.These equivalences provide the basis for corresponding equivalences of full DG norms.Source: §3.1 · p. 12, §3 · p. 12

3.2. Discrete Korn inequality

  • Discrete Korn inequalities yield equivalence of DG norms across the base discrete, discrete, and continuous surfaces.The discrete Korn result extends the continuous inequality to discrete functions that need not be tangentially conforming.Source: §3.2 · pp. 13, 16, 17

3.3. The BDM interpolant

  • The BDM interpolant is defined by mapping the standard affine-surface interpolant and satisfies a weak commuting property with stability and interpolation estimates.The usual commuting diagram property does not hold for the mapped interpolant.Source: §3.3 · pp. 17–18

3.4. Geometric Error Estimate

  • Geometric-error estimates control the variational crimes from posing the finite element method on the discrete surface rather than the continuous surface.The section treats bounds for piecewise smooth functions, functions in V_h, and smooth functions.Source: §3.4 · pp. 19, 21
  • The summed edge terms I_e, II_e, and III_e are bounded by h^2||u||H2(γ)||w||H2(γ).This completes the geometric-error estimate for the three edge contributions.Source: §3.4 · p. 28
  • The final edge bound assumes d(v1)-d(v0)=O(h^3) for the vertices of an edge.The proof combines this placement condition with a scaled trace inequality and other estimates.Source: §3.4 · p. 32

4. Finite element method and error analysis

This section formulates finite-element approximations for surface Stokes and establishes stability and optimal energy-norm error estimates. The error analysis assumes sufficiently small mesh size and a sufficiently large penalty parameter.

4. Finite element method and error analysis

  • The method section establishes optimal error estimates in the energy norm for finite-element approximations of surface Stokes.The stated analysis assumes sufficiently small h and sufficiently large penalty parameter ρ, independent of h and u.Source: §4 · p. 32, §4.4 · p. 37

4.1. Weak form and finite element method

  • For the weak formulation, the jump form vanishes on tangential componentwise H1 fields, reducing the form to the standard weak form.The formulation also holds for discrete test functions, by elementwise integration by parts.Source: §4.1 · p. 33

4.2. Discrete Inf-Sup Condition

  • For sufficiently small h, the discrete inf-sup condition holds with a constant independent of h.Source: §4.2 · p. 34

4.3. Coercivity and continuity

  • Coercivity requires a penalty parameter ρ sufficiently large independently of h; continuity of the form is also established.Source: §4.3 · pp. 34–35

4.4. Error analysis

  • The error theorem assumes u∈[H^{r+1}(γ)]^3 and p∈H^r(γ), with h sufficiently small and ρ sufficiently large.A stated estimate also requires k≥2, or k≥1 with neighboring vertex signed-distance differences bounded by O(h^3).Source: §4.4 · p. 37

5. Numerical examples

The numerical experiments confirm the predicted energy- and velocity-L2 convergence rates for isoparametric approximations, while lower-degree geometry can limit convergence. Tests also find optimal L2 velocity convergence in five lowest-order experiments that violate the proof’s mesh-node condition.

5.1. Numerical results

  • The numerical tests used a MATLAB implementation supporting arbitrary combinations of geometry and BDM degrees from 1 through 4, with a nonzero incompressibility constraint.The velocity was nonpolynomial because the projection operator was used.Source: §5.1 · p. 40
  • For isoparametric approximations k = r, energy error converges at h^r + h^k and velocity L2 error at h^(r+1) + h^(k+1), as predicted.Figure 1 shows these rates for r = 1, 2, 3, and 4.Source: §5.1 · p. 40, Figure 1 · p. 41
  • With geometry degree k = r −1, velocity H1 and pressure errors converge suboptimally at O(h^k), as Figure 2 confirms.For even k, the k = 2, r = 3 case initially converges near h^3 before tending to h^2; odd k follows h^k throughout.Source: §5.1 · p. 40, Figure 2 · p. 42
  • For k = 2 and r = 3 on the sphere, mesh nodes on the surface yield h^3 convergence, attributed to geometric superconvergence.The reported superconvergence is ∥d∥L∞(Γh,k) ≲ h^(k+1) when k is even; it may also explain the initially higher rate on the ellipsoid.Source: §5.1 · p. 41
  • Five lowest-order experiments retained optimal h^2 velocity L2 convergence despite edgewise signed-distance differences of approximately h^2, suggesting the proof condition may not be necessary.Each experiment violated the h^3-scale condition by perturbing nodes in different normal-direction patterns.Source: §5.1 · p. 41, Figure 3 · p. 42

Funding

Both authors were partially supported by NSF grant DMS-2012326.

Funding

  • Both authors were partially supported by NSF grant DMS-2012326.Source: Funding · p. 42
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