Surface Stokes Without Inf-Sup Condition

Mansur Shakipov; Ricardo H. Nochetto

REVIEW 2 major objections 4 minor 1 cited by

Surface Stokes Without Inf-Sup Condition

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A two-Laplacian reformulation of the surface Stokes problem eliminates the discrete inf-sup condition for arbitrary element pairs.

desk verdict A genuinely new elliptic reformulation of surface Stokes that sidesteps the discrete inf-sup condition, with clean analysis and convincing numerics — but the equivalence to the original Stokes problem holds only for data one to two orders more regular than the minimal setting. read the letter →

arxiv 2508.13342 v1 pith:GZCX3MK2 submitted 2025-08-18 math.NA cs.NAmath.AP

Ricardo H. Nochetto , Mansur Shakipov This is my paper · ORCID

classification math.NAcs.NAmath.AP MSC 35J4735J5035M3035Q3558J0565N1265N1565N30

keywords surfaceStokesellipticreformulationinf-supconditionparametricFEMWeingartenmapRiccicurvatureGårdinginequalityerrorestimates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the tangential Stokes problem on a closed $C^{3}$ hypersurface can be rewritten exactly as a nonsymmetric, indefinite elliptic system in which both the velocity and the pressure satisfy Laplace-type equations. For this two-Laplacian reformulation, a lifted parametric finite element method is well-posed for any combination of velocity and pressure polynomial degrees, provided the mesh size h is below a threshold h0 that depends only on the surface's Weingarten map. The paper proves quasi-best approximation in a mesh-dependent $H^{1}$ norm and optimal-order $L^{2}$ error estimates for both unknowns. The result matters because the discrete inf-sup condition, which normally forces specific element pairs or stabilization, is completely circumvented.

What carries the argument

The load-bearing identity is the decomposition of the Bochner Laplacian, $a(v, \nabla q) = (\mathrm{div}\, v, \Delta q) - (Rv, \nabla q)$ for $v \in H^1_t$ and $q \in H^2_\#$, with $R = \mathrm{tr}(W) W - W^2$ the Ricci curvature map. It converts the Stokes constraint into a pressure equation, producing the two-Laplacian system (3.3). The second engine is the argument for discrete well-posedness of indefinite Galerkin problems: the Gårding inequality leaves a compact term, which is absorbed using a conditional $L^2$ error estimate obtained by a duality argument; this yields a threshold $h_0$ that shrinks as the deviation of $R$ from the identity grows.

What would settle it

On the torus example of Section 5 with the same manufactured velocity but a pressure data $f$ chosen in $L^2_\# \setminus H^1_\#$, run the $P1/P1$ discretization of the reformulated system: if the discrete pressure still converges in $L^2$ to the true pressure, the regularity restriction in Remark 4.3 is unnecessary; if it fails to converge or converges to a different limit, the restriction is real.

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Extended reading notes

Core claim

The central claim is that the primitive-variable surface Stokes system (2.11) is equivalent to the elliptic system (3.3) on the space $V^{1,1} = H^1_t \times H^1_\#$, where the pressure is one order more regular than in the classical formulation. Testing the momentum equation with gradients of $H^2$ test functions and using the decomposition $a(v, \nabla q) = (\mathrm{div}\, v, \Delta q) - (Rv, \nabla q)$ turns the divergence constraint $\mathrm{div}\, u = f$ into a second Laplace equation for the pressure, yielding two coupled Laplace problems perturbed only by lower-order terms involving the Ricci curvature map $R = \mathrm{tr}(W) W - W^2$. The induced bilinear form is continuous, satisfies a Gårding inequality, and is an isomorphism on $V^{1,1}$. Consequently, for $h \le h_0$ with $h_0$ depending only on the Weingarten map, the lifted parametric FEM with arbitrary finite element spaces is uniquely solvable and satisfies $E_1^{h,1}[W,W_h] \le C_9 \inf_{V_h} E_1^{h,1}[W,V_h]$ and $\|W - W_{h,t}\|_{0,0} + h E_1^{h,1}[W,W_h] \le C_{11} h^2 \|G\|_{0,0}$.

Load-bearing premise

The load-bearing premise is that the data $F = (f, f)$ for the original Stokes problem lies in $H^{-1}_t(\mathrm{div}\,\Gamma) \times H^1_\#$, one order more regular in both the force divergence and the pressure than the minimal Stokes data space $H^{-1}_t \times L^2_\#$; if the pressure data is only $L^2$, the reformulated system's $H^1$ pressure may fail to recover the true pressure.

Editorial extensions

If this is right

Any velocity-pressure pair of polynomial degrees, equal or disparate, can be used without stabilizers or inf-sup-stable element choices, simplifying surface Stokes codes.
The method achieves optimal-order convergence in both variables: $\|W - W_{h,t}\|_{0,0} + h E_1^{h,1}[W,W_h] = O(h^2)$ for sufficiently smooth data, with no dependence of the convergence order on the element pair.
The same analysis covers the physically relevant Stokes model with the perturbed surface diffusion operator $\alpha I - \Delta_S$, $\alpha > 0$, in two dimensions.
The smallness condition $h \le h_0$ is tied to resolving the deviation of the Ricci curvature map from the identity on the tangent space, not to resolving principal curvatures.
The reformulation provides new high-regularity results for the Stokes solution when data lie in $H^{-1}_t(\mathrm{div}\,\Gamma) \times H^1_\#$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the pressure is sought in $H^1_\#$ instead of $L^2_\#$, the method trades the inf-sup condition for one extra order of data regularity; a natural next test is whether a pressure-stabilized variant could recover convergence for minimal-regularity data.
The two-Laplacian structure suggests block-diagonal preconditioners built from scalar Laplace-Beltrami solvers, which may be noticeably cheaper than standard Stokes preconditioners.
The same reformulation strategy could apply to other vector Laplace operators (for instance Hodge Laplacians) whenever a Ricci-type curvature identity relates the differential operators.
On evolving surfaces, discretizing the reformulated system in space and time might avoid inf-sup issues in the tangential Navier-Stokes setting, though this is not addressed in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuinely new idea with the proofs mostly in place. The authors rewrite the surface Stokes problem as an indefinite elliptic system driven by two Laplacians and show the bilinear form satisfies a Garding inequality, so any conforming finite element pair works without a discrete inf-sup condition once the mesh resolves the Ricci curvature. That is a real advance over the older stabilization/perturbation tricks, and the numerical experiments back it up, including equal-order pairs that are not inf-sup stable for the primitive formulation.

What is new and solid: the equivalence is demonstrated through an identity (Lemma 2.4) that is proved in the paper, not merely cited, so the companion [BNS25] is not load-bearing. The continuous well-posedness and the discrete analysis follow the Schatz–Aubin–Nitsche route cleanly. The numerics are honest: manufactured solutions, a nontrivial torus with sign-changing curvature, and rates that match the theory. The code is public.

The soft spots are real but not fatal. The main one, which the authors disclose in Remarks 4.3 and 4.13 but the abstract underplays, is that the reformulated elliptic problem is equivalent to the original Stokes problem only when the data lies in H^{-1}_t(divΓ) × H^1_#; the optimal L2 result needs H_t(divΓ) × H^2_#. This is one to two orders above the minimal V^{-1,0} data space. The stress-test example is correct: F=(∇p,0) with p in L^2_# but not H^1_# gives a perfectly valid Stokes solution, yet the reformulated data falls outside V^{-1,-1}, so the method as analyzed does not even have a well-defined right-hand side. The title should therefore be read with that regularity qualifier in mind. I would ask the authors to state this limitation more prominently, perhaps in the abstract, but it does not undermine the core construction.

Two smaller issues: Lemma 3.5 (adjoint H2 regularity) is asserted without proof and is needed for the duality estimate and the mesh threshold; it looks true, but it should be proved rather than hand-waved. And in Lemma 4.8 the constant C9 is printed with c_{A,1} in the denominator; the proof gives c_{A,1}^2. That is a typo, not a substantive error.

Bottom line: this paper deserves a serious referee and likely publication after the regularity qualification is made explicit and the missing adjoint-regularity proof is supplied. I would bring it to a reading group and would certainly cite it if I worked on surface Stokes.

Referee Report

2 major / 4 minor

Summary. The paper proposes a new formulation of the surface Stokes problem on a closed C^3 hypersurface, rewriting the saddle-point system (2.11) as a nonsymmetric indefinite elliptic system (3.3) of two Laplacians via the identity (2.12). The authors prove well-posedness of the continuous problem in V^{1,1} for data in V^{-1,-1}, derive a Garding inequality, and analyze a lifted parametric FEM with a weak tangentiality penalty. Following Schatz's approach, they prove discrete well-posedness for h <= h0 (depending only on the Weingarten map), a quasi-best approximation property in a mesh-dependent H^1 norm, and optimal L^2 error estimates for both velocity and pressure, for arbitrary polynomial degree pairs. Numerical experiments on a torus confirm the predicted convergence rates.

Significance. The paper's main contribution is a discretization of surface Stokes that provably avoids any discrete inf-sup condition: the elliptic reformulation (3.3) is well-posed with respect to the V^{1,1} norm, and the finite element analysis in Sections 4.7-4.11 delivers quasi-optimality and optimal L^2 rates for any velocity-pressure pair. The analysis is largely self-contained: the crucial identity (2.12) is proved in Lemma 2.4, well-posedness is connected to the standard Stokes problem, and the Schatz-style duality argument is carried out in detail. The numerical experiments are reproducible (code on GitHub) and show optimal orders, including for pairs like P1/P2 and P1/P3 that are not inf-sup stable for the original formulation. The main caveat is the data regularity required for the reformulation to be equivalent to Stokes; this should be prominently stated.

major comments (2)

[Abstract; §3.3 (Theorem 3.2); §4 (Remark 4.3)] The paper presents the method as a reformulation of the surface Stokes equations, but the equivalence and the discrete analysis hold only for Stokes data F=(f,f) with f in H^{-1}_t(divΓ) and f in H^1_# (for well-posedness) and f in H_t(divΓ), f in H^2_# (for the optimal L^2 estimate). Indeed, the elliptic system (3.3) is analyzed for G in V^{-1,-1}, whereas the data map D in (3.2) sends the minimal Stokes data space V^{-1,0} only to V^{-1,-2} (Lemma 3.1). For pressure data f merely in L^2_#, the second component g = -divΓ f - ΔΓ f lies in H^{-2}_# but not H^{-1}_#, so G[V_{h,t}] in (4.11) is not defined for V_{h,t} in V^{1,1}. The authors acknowledge this in Remark 4.3, but the abstract and introduction claim an unqualified reformulation of the surface Stokes equations. I recommend stating the data regularity in the abstract and introduction, and adding a brief discussion of how the method relates to, or could be extended for, the minimal-data setting.
[§3.4 (Lemma 3.5); used in §4.8 (Lemma 4.7)] Lemma 3.5 (higher regularity of the adjoint problem) is asserted without proof, with the note that the proof is similar to that of (3.9). This lemma is load-bearing: it provides the V^{2,2} regularity of the adjoint solution used in Lemma 4.7 to obtain the factor h in the L^2 error estimate, which in turn enters the definition of h0 in Lemma 4.8 and the final L^2 estimate in Theorem 4.11. Since the adjoint problem (3.14) has different signs and coupling than the forward problem, the proof is not literally identical to the forward case. Please supply the proof or a detailed sketch.

minor comments (4)

[§4.9 (Lemma 4.8)] The constant C9 is stated as C9 = 2 max{1,C_A^2}/c_{A,1}, but the proof (just before (4.19)) yields C9 = 2 max{1,C_A^2}/c_{A,1}^2. Please correct the statement.
[§3.5 (Proposition 3.6)] The stated continuity constant C_A = 1 + C_{P,t} max{1, ||R||_{0,∞}} is not consistent with the proof: the four terms give a bound of 2 + C_{P,t}(1 + ||R||_{0,∞}). Since the exact value is not used qualitatively, this is a minor correction, but the stated value should be valid.
[§4.6 (Lemma 4.5)] The derivation gives kh(I_h v, I_h v)^{1/2} = (1/τ)||n·I_h v|| ≤ (C/τ) h^2 ||v||_2. To conclude the stated O(h) bound with a constant independent of τ, one needs h ≲ τ (or the constant must be allowed to depend on τ). Please state this explicitly or record the τ-dependence of C7 and the subsequent constants.
[§6 (Conclusions)] Minor typos: 'parameteric' should be 'parametric'; the abstract has a missing space in 'C3'; consider using \bar{w} consistently for the pressure in (3.3)–(3.4) to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the elliptic reformulation is derived and proved in-paper, the discrete well-posedness is derived via Schatz duality, and numerical tests use standard manufactured solutions; self-citations are not load-bearing.

full rationale

The derivation chain is not circular. The key Bochner-Laplacian decomposition (2.12) is attributed to [BNS25, Lemma 2.1] but is then proved in Lemma 2.4 using [JOR18, Lemma 2.1] and the symmetry of the covariant Hessian, so the self-citation is not load-bearing for the central identity. Theorem 3.2 proves well-posedness of the elliptic reformulation by constructing data through the data-map isomorphism D (Lemma 3.1) and reducing to the standard Stokes problem (2.11), whose well-posedness rests on the Poincare inequality and surjectivity of the divergence; it does not assume the target discrete result. The discrete analysis (Lemmas 4.6-4.8 and Theorems 4.9-4.11) uses the Schatz/Aubin-Nitsche argument with Garding's inequality and the continuous solution as input, so the 'no discrete inf-sup' conclusion is derived rather than assumed. Numerical experiments use manufactured solutions with data computed from the exact solution, so no fitted parameter is renamed as a prediction. Self-citations to [BNS25] concern standard elliptic/Stokes regularity or an elementary Hessian symmetry; because the critical identity is proved in-paper and the remaining cited results are standard and parameter-free, they do not raise the circularity score. Flagged but non-circular: Lemma 3.5 states 'Its proof is very similar to the proof of (3.9) in Theorem 3.2 (elliptic reformulation) and will therefore be omitted'; this is a missing proof needed for the duality estimate (4.15) and for the threshold h0, and Remarks 4.3 and 4.13 explicitly narrow the method's scope to data F in H^{-1}_t(divGamma) x H^1_# (and H_t(divGamma) x H^2_# for optimal L2). Both are rigor or scope concerns, not circular reductions.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard elliptic and Stokes theory on smooth closed manifolds, on the vector Laplacian decomposition identity (2.12) proved in Lemma 2.4, and on standard finite element interpolation results. No new entities are introduced. The only free method parameter is the penalty coefficient tau, which is not fitted to data.

free parameters (2)

penalty parameter tau = not reported (tau <= 1)
Appears in (4.3) in the normal-penalty term k_tau(w,v) = tau^{-2} (w·n, v·n). The analysis holds for any tau <= 1, and the value in experiments is not stated.
mesh-size threshold h0 = h0 = sqrt(c_{A,1}/(4 c_{A,2} C8^2))
This is a derived threshold that depends on the Weingarten map through c_{A,2} and the duality constant C8. It is not fitted to data, but it restricts the mesh size in the main convergence theorems.

assumptions (8)

domain assumption Gamma is a compact, connected C^3 hypersurface without boundary, d >= 2.
Stated in Section 2 and used throughout; needed for H2 regularity of the Laplace-Beltrami and Bochner-Laplace operators.
standard math Poincare inequalities (2.7) hold on Gamma for scalar and tangential vector fields.
Section 2.3, cited from [DE13b; BDN20; HLL20; BNS25]; used to define equivalent H1 norms.
domain assumption Laplace-Beltrami and Bochner-Laplace problems have H2 regularity on C^3 surfaces.
Proposition 2.1, cited from [DE13b; BDN20; BNS25]; used in the data-map isomorphism and in the duality argument.
standard math The tangential divergence div_Gamma : H1_t -> L2_# is surjective.
Lemma 2.2, cited from [JOR18; BNS25]; ensures well-posedness of the continuous Stokes problem.
standard math Identity (2.13): P div_Gamma grad_Gamma^T v = grad_Gamma div_Gamma v + R v.
Cited from [JOR18, Lemma 2.1]; used in the proof of Lemma 2.4, the central decomposition identity.
standard math The covariant Hessian of a scalar function is symmetric.
Used in Lemma 2.4; cited from [BNS25, Proposition C.5].
domain assumption Korn's inequality for the perturbed surface diffusion operator alpha I - Delta_S for d = 2.
Used in Section 3.7 to establish well-posedness of the perturbed surface diffusion Stokes problem; cited from [JOR18, Lemma 4.1].
domain assumption The discrete mesh is shape-regular and quasi-uniform, and h is small enough for the closest point projection to be bijective.
Section 4.1; standard assumptions for lifted parametric FEM.

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Pith. "Pith review of Surface Stokes Without Inf-Sup Condition." pith.science (2026). https://pith.science/paper/GZCX3MK2

@misc{pith2026250813342,
  author       = {Pith},
  title        = {Pith review of: Surface Stokes Without Inf-Sup Condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZCX3MK2}},
  note         = {Machine review of arXiv:2508.13342}
}

abstract

For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis for a numerical method based on lifted parametric FEM. Assuming no geometric error for simplicity, we prove its well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as an optimal $L^2$ error estimate for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. We present numerical experiments for velocity-pressure pairs with equal and disparate polynomial degrees, demonstrating that the proposed method is both accurate and practical.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Stabilized Morley FEM for surface Stokes in stream-function formulation: Optimal convergence via a new geometric estimate
math.NA 2026-07 accept novelty 6.0 of 10

A parameter-free stabilized Morley method for surface Stokes in stream-function form achieves first-order broken-H2 and second-order broken-H1 convergence under H3 regularity, via a new normal-separated geometric estimate.

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Works this paper leans on

13 extracted references · 11 canonical work pages · cited by 1 Pith paper

[1]

Relaxation dynamics of fluid membranes

[AD09] M. Arroyo and A. DeSimone. “Relaxation dynamics of fluid membranes”. In: Phys. Rev. E 79 (3 Mar. 2009). [BMN02] E. B ¨ansch, P. Morin, and R. H. Nochetto. “An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition”. In: SIAM J. Numer. Anal.40.4 (2002). [BNS25] G. A. Benavides, R. H. Nochetto, and M. Shakipov. Lp-based ...

work page 2002
[2]

A posteriori error estimates for the Laplace-Beltrami operator on parametric C 2 surfaces

Experimental order of convergence for non-inf-sup-stable velocity- pressure pairs for (1.1) (with pressure having the higher polynomial degree). Here, ℓ≥ 1 is the refinement level. The rates are either consistent with theory or better. [BD19] A. Bonito and A. Demlow. “A posteriori error estimates for the Laplace-Beltrami operator on parametric C 2 surface...

work page 2019
[7]

Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes prob- lem

arXiv: 2506.20419 [math.NA]. [DN89] R. G. Dur ´an and R. H. Nochetto. “Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes prob- lem”. In: SIAM J. Numer. Anal.26.6 (1989). [Dzi88] G. Dziuk. “Finite elements for the Beltrami operator on arbitrary surfaces”. In: Partial differential equations and calculus of variations. V ol

arXiv 1989
[9]

SFEM for the Lagrangian formulation of the surface Stokes problem

arXiv: 2410.19470 [math.NA]. [EG04] A. Ern and J.-L. Guermond. Theory and practice of finite elements. V ol

work page Pith review arXiv
[1984]

Stabilized mixed methods for the Stokes problem

[BD88] F. Brezzi and J. Douglas Jr. “Stabilized mixed methods for the Stokes problem”. In: Numer. Math. 53.1-2 (1988). [Bur+18] E. Burman, P. Hansbo, M. G. Larson, and A. Massing. “Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions”. In: ESAIM Math. Model. Numer. Anal. 52.6 (2018). 20 REFERENCES [...

work page 1988
[1988]

L2-estimates for the evolving surface finite element method

[DE13a] G. Dziuk and C. M. Elliott. “ L2-estimates for the evolving surface finite element method”. In: Math. Comp. 82.281 (2013). [DE13b] G. Dziuk and C. M. Elliott. “Finite element methods for surface PDEs”. In: Acta Numer. 22 (2013). [EM25] C. M. Elliott and A. Mavrakis. SFEM for the Lagrangian formulation of the surface Stokes problem

work page 2013
[2004]

Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds

[Fri18] T.-P. Fries. “Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds”. In: Internat. J. Numer. Methods Fluids 88.2 (2018). [GLR18] J. Grande, C. Lehrenfeld, and A. Reusken. “Analysis of a high-order trace finite element method for PDEs on level set surfaces”. In: SIAM J. Numer. Anal.56.1 (2018). [HLL20] P. Hansbo, M. G. Larso...

work page 2018
[2009]

Finite element interpolation of nonsmooth functions satisfying boundary conditions

[SZ90] L. R. Scott and S. Zhang. “Finite element interpolation of nonsmooth functions satisfying boundary conditions”. In: Math. Comp. 54.190 (1990). [Scr60] L. Scriven. “Dynamics of a fluid interface Equation of motion for Newtonian surface fluids”. In: Chemical Engineering Science 12.2 (1960). [SS13] J. Slattery and L. Sagis. Interfacial Transport Pheno...

work page 1990

Show all 13 references

[2011]

On the stabilization of finite element approximations of the Stokes equations

[BP84] F. Brezzi and J. Pitk ¨aranta. “On the stabilization of finite element approximations of the Stokes equations”. In:Efficient solutions of elliptic systems (Kiel, 1984). V ol

1984
[2013]

Stokes flow of an evolving fluid film with arbitrary shape and topology

[ZSC25] C. Zhu, D. Saintillan, and A. Chern. “Stokes flow of an evolving fluid film with arbitrary shape and topology”. In: Journal of Fluid Mechanics 1003 (2025). DEPARTMENT OF MATHEMATICS , UNIVERSITY OF MARYLAND , COLLEGE PARK , MD 20742 Email address: {rhn,shakipov}@umd.edu

2025
[2017]

A finite element method for elliptic equations on surfaces

[ORG09] M. A. Olshanskii, A. Reusken, and J. Grande. “A finite element method for elliptic equations on surfaces”. In: SIAM J. Numer. Anal.47.5 (2009). [ORS24] M. A. Olshanskii, A. Reusken, and P. Schwering. “An Eulerian finite element method for tangential Navier-Stokes equat...

2009
[2020]

Finite element discretization methods for velocity- pressure and stream function formulations of surface Stokes equations

[Bra+22] P. Brandner, T. Jankuhn, S. Praetorius, A. Reusken, and A. V oigt. “Finite element discretization methods for velocity- pressure and stream function formulations of surface Stokes equations”. In: SIAM J. Sci. Comput. 44.4 (2022). [BR20] P. Brandner and A. Reusken. “Fi...

2022
[2025]

[BBF13] D

arXiv: 2508.11109 [math.AP]. [BBF13] D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications. V ol

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Relaxation dynamics of fluid membranes

[AD09] M. Arroyo and A. DeSimone. “Relaxation dynamics of fluid membranes”. In: Phys. Rev. E 79 (3 Mar. 2009). [BMN02] E. B ¨ansch, P. Morin, and R. H. Nochetto. “An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition”. In: SIAM J. Numer. Anal.40.4 (2002). [BNS25] G. A. Benavides, R. H. Nochetto, and M. Shakipov. Lp-based ...

work page 2002

[2] [2]

A posteriori error estimates for the Laplace-Beltrami operator on parametric C 2 surfaces

Experimental order of convergence for non-inf-sup-stable velocity- pressure pairs for (1.1) (with pressure having the higher polynomial degree). Here, ℓ≥ 1 is the refinement level. The rates are either consistent with theory or better. [BD19] A. Bonito and A. Demlow. “A posteriori error estimates for the Laplace-Beltrami operator on parametric C 2 surface...

work page 2019

[3] [7]

Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes prob- lem

arXiv: 2506.20419 [math.NA]. [DN89] R. G. Dur ´an and R. H. Nochetto. “Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes prob- lem”. In: SIAM J. Numer. Anal.26.6 (1989). [Dzi88] G. Dziuk. “Finite elements for the Beltrami operator on arbitrary surfaces”. In: Partial differential equations and calculus of variations. V ol

arXiv 1989

[4] [9]

SFEM for the Lagrangian formulation of the surface Stokes problem

arXiv: 2410.19470 [math.NA]. [EG04] A. Ern and J.-L. Guermond. Theory and practice of finite elements. V ol

work page Pith review arXiv

[5] [1984]

Stabilized mixed methods for the Stokes problem

[BD88] F. Brezzi and J. Douglas Jr. “Stabilized mixed methods for the Stokes problem”. In: Numer. Math. 53.1-2 (1988). [Bur+18] E. Burman, P. Hansbo, M. G. Larson, and A. Massing. “Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions”. In: ESAIM Math. Model. Numer. Anal. 52.6 (2018). 20 REFERENCES [...

work page 1988

[6] [1988]

L2-estimates for the evolving surface finite element method

[DE13a] G. Dziuk and C. M. Elliott. “ L2-estimates for the evolving surface finite element method”. In: Math. Comp. 82.281 (2013). [DE13b] G. Dziuk and C. M. Elliott. “Finite element methods for surface PDEs”. In: Acta Numer. 22 (2013). [EM25] C. M. Elliott and A. Mavrakis. SFEM for the Lagrangian formulation of the surface Stokes problem

work page 2013

[7] [2004]

Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds

[Fri18] T.-P. Fries. “Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds”. In: Internat. J. Numer. Methods Fluids 88.2 (2018). [GLR18] J. Grande, C. Lehrenfeld, and A. Reusken. “Analysis of a high-order trace finite element method for PDEs on level set surfaces”. In: SIAM J. Numer. Anal.56.1 (2018). [HLL20] P. Hansbo, M. G. Larso...

work page 2018

[8] [2009]

Finite element interpolation of nonsmooth functions satisfying boundary conditions

[SZ90] L. R. Scott and S. Zhang. “Finite element interpolation of nonsmooth functions satisfying boundary conditions”. In: Math. Comp. 54.190 (1990). [Scr60] L. Scriven. “Dynamics of a fluid interface Equation of motion for Newtonian surface fluids”. In: Chemical Engineering Science 12.2 (1960). [SS13] J. Slattery and L. Sagis. Interfacial Transport Pheno...

work page 1990