A Nitsche method for incompressible fluids with general dynamic boundary conditions

Erika Maringov\'a Kokavcov\'a; Franz Gmeineder; Pablo Alexei Gazca-Orozco; Tabea Tscherpel

REVIEW 2 major objections 4 minor 1 cited by

A Nitsche method for incompressible fluids with general dynamic boundary conditions

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Nitsche scheme now covers general slip laws at fluid walls, with convergence to weak solutions.

desk verdict Substantial and novel convergence framework for Nitsche slip-boundary discretizations, but the pressure compactness step in Lemma 5.13 is a genuine gap that needs repair before the main theorem is sound. read the letter →

arxiv 2502.09550 v2 pith:OBELVMMT submitted 2025-02-13 math.NA cs.NA

Pablo Alexei Gazca-Orozco , Franz Gmeineder , Erika Maringová Kokavcová , Tabea Tscherpel This is my paper · ORCID

classification math.NAcs.NA MSC 65N3076D0776M10

keywords NitschemethoddynamicboundaryconditionsslipNavier-StokesequationsKorninequalityTrescafrictionweaksolutionsfiniteelementconvergence

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

The paper introduces a fully discrete finite element scheme for the time-dependent incompressible Navier-Stokes equations on polyhedral domains, imposing the impermeability condition weakly through a Nitsche penalty rather than enforcing it exactly. It handles an unusually broad family of slip boundary conditions, including Navier slip, perfect slip, Tresca friction, stick-slip, and dynamic, set-valued, non-monotone and non-coercive relations. The main claim is that, as the mesh size, time step, and regularisation parameter simultaneously tend to zero, subsequences of discrete solutions converge to a weak solution of the problem, thereby also proving existence of weak solutions in settings not previously covered. A sympathetic reader should care because boundary slip is itself a constitutive law of the fluid-solid interface, and a single numerical framework that provably converges for such general laws fills a real gap.

What carries the argument

The load-bearing ingredient is a Korn-type inequality with a normal trace term (Theorem 3.2), which bounds the full H1 norm of a velocity field by the L2 norm of its symmetric gradient plus the L2 norm of the normal trace on the boundary. This inequality compensates for the fact that Nitsche penalisation does not impose u·n = 0 strongly, and it is combined with an inverse trace inequality to absorb the non-monotonicity parameter λ through a sharp trace constant. For implicit relations, a generalised Yosida regularisation of the monotone graph, together with a Minty-type convergence lemma, identifies the nonlinear boundary term in the limit; for r ≤ 2, strong convergence of tangential traces makes the identification more direct.

What would settle it

Construct a polyhedral domain and an explicit boundary law s(v) = −γ v with γ larger than the threshold 2ν/c_tr,K² (an 'active wall' type relation), and run the proposed Nitsche scheme with λ = γ on a sequence of meshes. The theorem predicts that the energy estimate (5.26) cannot be closed, so a concrete observation would be that discrete norms fail to stay bounded or that the computed solutions fail to converge as h, δ, ε → 0; conversely, if they still converge, the threshold is not necessary and the assumption could be relaxed.

Watch

Extended reading notes

Core claim

The paper proves that a symmetric Nitsche method with backward Euler time stepping converges, in the simultaneous limit of discretisation and regularisation parameters, to a weak solution of the Navier-Stokes equations with impermeability and the general dynamic slip condition (2.14), under either an explicit noncoercive relation (2.15) or an implicit, possibly non-monotone coercive relation (2.16). For explicit noncoercive relations with exponent r in [1,2], Theorem 5.5 establishes convergence; for implicit relations with r in (2,2♯), an antisymmetric variant (5.123) achieves the same via Theorem 5.19. The proof also establishes existence of weak solutions, extending earlier existence results to the r=1, noncoercive, and non-monotone cases.

Load-bearing premise

The proof requires the non-monotonicity parameter λ to be strictly below 2ν/c_tr,K², so that the negative boundary term −λ∥tr_τ(u)∥²_{L²(Γ)} can be absorbed by the coercivity of the Nitsche/Korn norm; if the slip law is too non-monotone, the a priori estimates do not close and convergence is not claimed.

Editorial extensions

If this is right

If the convergence theorem is correct, it supplies the first Nitsche-based numerical scheme for fluid equations with such general slip and dynamic boundary conditions, covering Tresca, stick-slip, power-law, and non-monotone laws in one framework.
The simultaneous limit argument gives a constructive proof of existence of weak solutions for the underlying PDE, including cases previously open such as r = 1, noncoercive relations, and non-monotone implicit relations with small λ.
The same machinery extends to mixed boundary conditions, with slip on part of the boundary and Dirichlet, periodic, or suitably handled natural conditions on the rest, as noted in Remark 5.6.
The numerical experiments indicate the method captures the relaxation behaviour of dynamic slip laws, meaning non-monotone-in-time tangential velocity, which experimental and analytic studies associate with polymer melts.
The pressure-free formulation and the Korn inequality with normal traces are reusable tools for other problems where boundary conditions are weakly imposed and tangential data are uncontrolled.

Reading between the lines

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

A natural testable extension is to replace the polyhedral domain by a curved one approximated by polyhedra; the authors indicate this as future work, and the Korn inequality with normal traces is already aligned with such a setting, though the convergence proof would need additional geometric stability estimates.
The λ-threshold condition (5.11) suggests that the method's practical range for non-monotone slip laws is limited by the sharp trace constant c_tr,K; choosing locally adaptive penalty parameters, as mentioned for monotone relations, could mitigate the restriction but is not covered by the present proof.
The framework is restricted to Newtonian fluids, but the boundary treatment is largely independent of the bulk constitutive law; one could test whether the same Nitsche approach extends to power-law or other non-Newtonian bulk models with the same boundary conditions.
The convergence result is qualitative; quantifying rates under stronger regularity assumptions, and examining how the observed relaxation behaviour depends on β, λ, and the regularisation parameter ε, would be a concrete next step for applications.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

Two things to know about arXiv:2502.09550. First, it is a real advance: a Nitsche method for Navier–Stokes with a genuinely broad class of slip boundary conditions—set-valued, nonmonotone, noncoercive, dynamic—with a convergence proof to weak solutions. That is new, and the unified treatment is well executed. The inhomogeneous Korn inequality with normal trace terms in Section 3 is a nice standalone result. Second, there is a load-bearing gap in the pressure compactness argument. Lemma 5.13 defines the time primitive of the discrete pressure and claims it is bounded in L^{8/(d+4)}(I; L²) by Lemma 5.11. Lemma 5.11 only gives δ² Σ ||π_j||^{8/(d+4)}_{L²} ≤ c. That does not control the primitive. Take any fixed mean-zero q and set π_j = δ^{-2/p} q with p = 8/(d+4); the δ² estimate holds, but the L^p norm of the primitive diverges. Since the distributional limit ξ is the only route to the pressure term in the limiting equation, the main theorem is not established as written. The fix is likely straightforward—sum the discrete pressure equation over time and bound the cumulative pressure directly—but it is absent from the manuscript.

The rest is in much better shape. The a priori estimates, compactness, trace identification for r ≤ 2, and the Minty argument for r > 2 are standard but carefully done. The numerical experiments are illustrative rather than a rigorous verification of convergence rates, which is acceptable for an existence paper. The practical restriction λ < 2ν/c²_{tr,K} is stated, but no guidance is offered for checking it for a given slip law; that is a real but minor limitation.

I think the paper deserves a serious referee. The gap in Lemma 5.13 should be flagged, and the authors should be asked to supply the missing pressure argument before acceptance. The contribution is significant enough that a repair is worth requesting rather than rejecting outright. I would not cite the convergence theorem as proven until the gap is closed, but I would cite the Korn inequality and the framework. My recommendation: send it to review with a request for a concrete fix on Lemma 5.13.

Referee Report

2 major / 4 minor

Summary. The paper proposes a fully discrete finite element scheme for the time-dependent Navier–Stokes equations on polyhedral domains with impermeability and a very general class of dynamic, possibly set-valued, nonmonotone, noncoercive, or non-smooth slip boundary conditions. The scheme combines an inf-sup stable mixed finite element pair, backward Euler time stepping, a Nitsche penalisation of the normal velocity, and a regularisation of implicit boundary relations. The main results, Theorems 5.5 and 5.19, claim subsequential convergence of the discrete solutions, in the simultaneous limit of mesh size, time step, and regularisation parameter, to a weak solution of the continuous problem, thereby also establishing existence. The proof proceeds through Korn-type inequalities with normal trace terms, a priori estimates, discrete Aubin–Lions compactness, convergence of the pressure via a primitive variable, identification of nonlinear boundary terms, and a separate antisymmetric Nitsche method for exponents r>2. Numerical experiments illustrate the behaviour of the method for nonmonotone slip, Tresca/stick-slip laws, and dynamic boundary conditions.

Significance. If the central convergence claim is established, the paper is a significant contribution to the numerical analysis of incompressible flows with general slip boundary conditions. It unifies many previously treated boundary laws, covers the set-valued and r=1 cases without variational inequalities, handles noncoercive and nonmonotone relations under quantitative smallness assumptions on the nonmonotonicity, and provides the first Nitsche-type analysis for such a broad class. The paper also proves a self-contained Korn inequality with normal trace terms, which is of independent interest. The proof is largely built on explicit lemmas rather than external black boxes, and the numerical experiments are accompanied by archived code. The main weakness is that one of the interpolation tools used in the limit passage is not correctly stated, and one pressure-compactness bound is asserted without proof; both are repairable but currently leave gaps in the proof of the main theorems.

major comments (2)

[Section 4.3.1, Lemma 4.11(b)] Lemma 4.11(b) and its proof are not correct as stated. For P1 finite element functions on a polyhedral domain with Γ = ∂Ω, the condition v_h·n = 0 on all boundary faces forces the nodal value at any boundary vertex lying on faces with non-collinear normals to be zero; consequently L¹_1(Th)^d ∩ H¹_n(Ω)^d does not contain functions with arbitrary tangential boundary traces, and no operator mapping into L¹_1(Th)^d that is a projection onto that space can satisfy (b) for all v ∈ H¹_n. The identity (I_h v)·n = I_h(v·n) in the proof is not justified, since n is not a scalar multiplier with which the volume Scott–Zhang operator commutes. This matters because Proposition 5.14 tests the discrete equation with v_h = I_h v satisfying v_h·n = 0, and uses this to drop the pressure and Nitsche boundary terms in (5.99)–(5.109). The authors need either to construct a higher-order trace-preserving interpolation operator whose boundary degrees of freedom are compatible with the subspace H¹_n, or to handle the boundary terms involving π^k and u^k·n by a different argument.
[Section 5.4, Lemma 5.13] The bound ||ξ_k||_{L^{8/(d+4)}(I;L²)} ≤ c is asserted to follow from Lemma 5.11 without proof. The proposed counterexample with π_j = δ^{-2/p} q does not satisfy the hypothesis of Lemma 5.11, since δ² Σ_j ||π_j||^p is then of order δ^{-1}, so it does not refute the lemma. Nevertheless, the implication is not immediate and it is load-bearing: Lemma 5.13 is the only source of the weak limit ξ used in the pressure term of (5.94). The estimate can be proved using the discrete Hardy inequality in l^p for p = 8/(d+4) > 1, which gives a bound of order δ^{(p-1)/p}; the manuscript should supply this argument in the proof of Lemma 5.13.

minor comments (4)

[Section 5.4, Lemma 5.12] The proof contains two consecutive steps both labelled "2. Step"; the second should be renumbered as Step 4.
[Remark 5.7] In the sentence "Q := I × Q", the same letter Q is used for the parabolic cylinder and presumably for the spatial domain; the notation should be disambiguated.
[Section 4.3.1, Lemma 4.11] The statement says that I_h is a projection onto L¹_1(Th)^d, but the proof does not explain how such a projection can also enforce the condition tr(I_h v)·n = 0 for all v ∈ H¹_n. Even after replacing L¹_1 with a higher-order space, the authors should state precisely in what sense the operator is a projection.
[Example 4.3] The verification of Assumption 4.1(a4) for the regularised Tresca relation is quite terse, especially the passage involving the Cayley transform and the indicator-type set S; a slightly more detailed explanation would help the reader verify the construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proof is self-contained; prior self-citations are auxiliary tools, not load-bearing inputs.

full rationale

The paper's central claim is that subsequences of the fully discrete Nitsche solutions converge to a weak solution of the Navier-Stokes system with general dynamic slip boundary conditions. The derivation chain is built in the manuscript: discrete a priori estimates (Lemma 5.8), time-increment and pressure estimates (Lemmas 5.10 and 5.11), compactness by a discrete Aubin-Lions lemma (Lemma 4.14), limit passage in the discrete equation (Proposition 5.14), and identification of the nonlinear boundary relation via strong convergence or Minty-type monotonicity arguments (Lemmas 5.16, 5.17 and 5.20). None of these steps defines the target weak solution in terms of the discrete scheme or conversely; the limiting object is characterized by the original PDE formulation independently of the numerical construction. The cited works by the same authors, notably ABM21, BMM23 and Tsc18, are used as context, as sources of modeling conventions, or as known auxiliary lemmas such as generalized Yosida approximation properties. In the places where a Minty-type convergence statement is needed, the paper states and proves Lemma 4.5 rather than simply importing the conclusion; the citation to Tsc18 concerns a technical regularisation property, not the convergence theorem itself. Thus the self-citations are not load-bearing in the sense of forcing the main result by authority. The pressure-compactness issue suggested by the skeptic, concerning whether the bound in Lemma 5.13 follows from Lemma 5.11, would, if valid, be a correctness gap in the proof rather than a circular reduction of the conclusion to an input; it therefore does not affect the circularity score. Overall, the derivation is self-contained against external benchmarks and no step reduces by construction to its own assumptions.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the geometric Korn inequality, the smallness condition on lambda, and the existence of regularising approximations for implicit slip laws. These are stated as assumptions rather than derived. No new physical entities are introduced; the auxiliary variable sigma is the tangential wall shear stress.

free parameters (2)

Nitsche penalty parameter alpha = alpha = 10 in experiments; alpha sufficiently large in theory (Assumption 5.4)
Controls weak enforcement of impermeability. The convergence theorem only requires alpha to exceed a constant depending on d, c_tr, and lambda; the numerical value is a method choice, not fitted to data.
Regularisation parameter epsilon = epsilon = 0.0002 in experiments; epsilon tends to 0 in theory
Replaces set-valued slip relations by smooth approximations. The theorem is asymptotic in epsilon and does not depend on a particular value.

assumptions (5)

domain assumption Assumption 3.1: v maps to ||v.n||_{L^q(Gamma)} is a norm on the space of rigid deformations R(Omega)
Essential for the Korn-type inequality (Theorem 3.2). It holds for polyhedral Gamma = boundary Omega with two non-collinear face normals (Corollary 3.7), but fails for axisymmetric domains and balls (Example 3.6).
ad hoc to paper Assumption 5.4: lambda < 2*nu/c_tr,K^2 and alpha > 2*(c_lambda*d*c_tr^2 + lambda)/(c_lambda - lambda)
Provides coercivity of the discrete energy (Lemma 5.8). The theorem is explicitly conditional on this smallness condition on the non-monotone slope lambda.
ad hoc to paper Assumption 4.1: existence of monotone regularisations s_epsilon satisfying (a1)-(a5), including strong convergence s_epsilon(v_epsilon) to s
Needed to approximate implicit set-valued relations. Generalised Yosida approximations are constructed for r>1, and a special construction for r=1; property (a5) is required for the Minty lemma in the r>2 case.
domain assumption Assumption 2.2 (A4): the zero set of g is maximal monotone as a subset of L^r x L^{r'}, guaranteed for r=1 by the convex potential condition (A5)
Used in Lemmas 4.4 and 4.5 to identify the limit relation; verified for Tresca and stick-slip examples.
standard math Standard background results: Necas Korn inequality, Gallouet-Latche discrete Aubin-Lions lemma, Brouwer fixed point theorem, discrete inf-sup condition for the chosen finite element pairs
Tools invoked in the convergence proof; the paper states the versions it uses.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Nitsche method for incompressible fluids with general dynamic boundary conditions." pith.science (2026). https://pith.science/paper/OBELVMMT

@misc{pith2026250209550,
  author       = {Pith},
  title        = {Pith review of: A Nitsche method for incompressible fluids with general dynamic boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBELVMMT}},
  note         = {Machine review of arXiv:2502.09550}
}

read the original abstract

Both Newtonian and non-Newtonian fluids may exhibit complex slip behaviour at the boundary. We examine a broad class of slip boundary conditions that generalises the commonly used Navier slip, perfect slip, stick-slip and Tresca friction boundary conditions. In particular, set-valued, nonmonotone, noncoercive and dynamic relations may occur. For a unifying framework of such relations, we present a fully discrete numerical scheme for the time-dependent Navier-Stokes equations subject to impermeability and general slip-type boundary conditions on polyhedral domains. Based on compactness arguments, we prove convergence of subsequences, finally ensuring the existence of a weak solution. The numerical scheme uses a general inf-sup stable pair of finite element spaces for the velocity and pressure, a regularisation approach for the implicit slip boundary condition and, most importantly, a general Nitsche method to impose the impermeability and a backward Euler time stepping. One of the key tools in the convergence proof is an inhomogeneous Korn inequality that includes a normal trace term.

Figures

Figures reproduced from arXiv: 2502.09550 by the authors.

**Figure 1.** Examples of relations covered by the framework in Section 2.1. Example 2.8 (implicit, non-monotone). Also non-monotone and implicit relations are covered. The following example is presented in [FCHCD20], where it is handled by means of variational inequalities: σ = µ(|uτ |) uτ |uτ | if uτ ̸= 0 |σ| ≤ µ(0) if uτ = 0 where µ(s) := (a − b)e −αs + b, for s ≥ 0, (2.21) for given constants a > b ≥ 0 and α ≥ 0. For λ := α(a… view at source ↗

**Figure 2.** Assumption 3.1 and its geometric impact. Left-hand figure: An axisymmetric cone Ω, for which Ax⊥n(x) holds for any x ∈ ∂Ω, where Ax := x × ξ. Note that n(x) is always contained in the plane spanned by ξ and x, and Ax is orthogonal to this plane; see Example 3.6. Right-hand figure: Polyhedral domains as the overall setting of the paper, and Corollary 3.7. If Γ ⊂ ∂Ω is polyhedral and contains two non-collinear normals… view at source ↗

**Figure 3.** Exact (red) and computed (blue) constitutive relation on Γs for the smooth relation (6.1). (a) Λ⋆ = 1. (b) Λ⋆ = 10 [PITH_FULL_IMAGE:figures/full_fig_p049_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Wall stress and tangential velocity on Γs for the smooth relation (6.1). (a) Λ⋆ = 0.6. (b) Λ⋆ = 5 [PITH_FULL_IMAGE:figures/full_fig_p049_4.png]

**Figure 5.** Figure 5: Exact (red) and computed (blue) constitutive relation on Γs for the non-smooth problem (6.4). the non-smooth problem with (6.4) are shown in Figures 5 and 6. The results match with the behaviour expected from the model [PITH_FULL_IMAGE:figures/full_fig_p049_5.png]

**Figure 6.** Figure 6: Wall stress and tangential velocity on Γs for the non-smooth problem (6.4). 6.2. Tresca and stick-slip. Let us consider the unsteady problem with stick-slip or Tresca boundary condition, see Example 2.7, given by σ = γ⋆uτ + µ⋆ uτ |uτ | if uτ ̸= 0 |σ| ≤ µ⋆ if uτ = 0 fo…

**Figure 7.** Figure 7: Wall stress and tangential velocity on ΓS for the unsteady stick-slip condition (6.7). Acknowledgements. We thank Ridgway Scott for helpful discussions on the Scott–Zhang interpolation operator, see Lemma 4.11. References [AA99] G. Alberti and L. Ambrosio. “A geometri…

**Figure 8.** Figure 8: Exact (red) and computed (blue) constitutive relation on ΓS for the unsteady stick-slip condition (6.7) [PITH_FULL_IMAGE:figures/full_fig_p052_8.png]

**Figure 9.** Figure 9: Time-dependence of the slip velocity for the dynamic boundary condition (6.9) at (x, y) = (0.5, 1). [Bab59] I. Babuška. “Die Abhängigkeit der Lösung der Elastizitätsprobleme von kleinen Veränderungen des Definitionsgebietes”. ZAMM - Journal of Applied Mathematics and …

Discussion (0). Continue with ORCID to comment.

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

A local Fortin projection for the Scott-Vogelius elements on general meshes
math.NA 2025-12 conditional novelty 7.0 of 10

A local Fortin projection for Scott-Vogelius k>=4 elements is constructed on general 2D shape-regular triangulations, including singular vertices, with divergence, trace, and stability properties.

Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages · cited by 1 Pith paper

[273]

Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition

doi: 10.1515/jnma-2023-0028. [BD99] E. Bänsch and K. Deckelnick. “Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition”.M2AN Math. Model. Numer. Anal.33.5 (1999), pp. 923–938. doi: 10.1051/m2an:1999126. [BDG20] D. Breit, L. Diening, and F. Gmeineder. “On the trace operator for functions of bounded A-variation”. A...

work page arXiv 1999
[773]

doi: 10.1007/s00211-023-01348-w. (P. A. Gazca-Orozco)Department of Applied Mathematics, University of Freiburg, 79104, Freiburg, Germany Email address: alexei.gazca@mathematik.uni-freiburg.de (F. Gmeineder) F achbereich Mathematik und Statistik, Universität Konstanz, Universitätsstr. 10, 78464 Konstanz, Germany. Email address: franz.gmeineder@uni-konstanz...

work page doi:10.1007/s00211-023-01348-w
[849]

On some techniques for approximating boundary conditions in the finite element method

doi: 10.1093/imanum/dry097. 58 REFERENCES [Ste95] R. Stenberg. “On some techniques for approximating boundary conditions in the finite element method”. Vol. 63. 1-3. International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994). 1995, pp. 139–148.doi: 10.1016/ 0377-0427(95)00057-7. [Sto45] G. G. Stokes. “On the th...

arXiv 1973
[2023]

Wallslipofmoltenpolymers

doi: 10.25561/104839. [Hat12] S.G.Hatzikiriakos.“Wallslipofmoltenpolymers”. Progress in Polymer Science37.4(2012). Topical Issue on Polymer Physics, pp. 624–643. doi: https : / / doi . org / 10 . 1016 / j . progpolymsci.2011.09.004. [HCJ21] W. Han, K. Czuprynski, and F. Jing. “Mixed finite element method for a hemivariational inequality of stationary Navi...

work page arXiv 2012
[2212]

A fully asynchronous mul- tifrontal solver using distributed dynamic scheduling

doi: 10.1142/S0218202521500470. [ADLK01] P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, and J. Koster. “A fully asynchronous mul- tifrontal solver using distributed dynamic scheduling”.SIAM J. Matrix Anal. Appl.23.1 (2001), pp. 15–41.doi: 10.1137/S0895479899358194. [AZ24] R. Altmann and C. Zimmer. “A second-order bulk-surface splitting for parabolic proble...

work page arXiv 2001
[5386]

Monotone operators in divergence form with x- dependent multivalued graphs

doi: 10.3934/dcds.2016036. [FMT04] G. Francfort, F. Murat, and L. Tatar. “Monotone operators in divergence form with x- dependent multivalued graphs”.Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)7-B.1 (Feb. 2004), pp. 23–59. [FS95] J. Freund and R. Stenberg. “On weakly imposed boundary conditions for second order problems”. Finite elements in fluids...

work page doi:10.3934/dcds.2016036 2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [273]

Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition

doi: 10.1515/jnma-2023-0028. [BD99] E. Bänsch and K. Deckelnick. “Optimal error estimates for the Stokes and Navier–Stokes equations with slip-boundary condition”.M2AN Math. Model. Numer. Anal.33.5 (1999), pp. 923–938. doi: 10.1051/m2an:1999126. [BDG20] D. Breit, L. Diening, and F. Gmeineder. “On the trace operator for functions of bounded A-variation”. A...

work page arXiv 1999

[2] [773]

doi: 10.1007/s00211-023-01348-w. (P. A. Gazca-Orozco)Department of Applied Mathematics, University of Freiburg, 79104, Freiburg, Germany Email address: alexei.gazca@mathematik.uni-freiburg.de (F. Gmeineder) F achbereich Mathematik und Statistik, Universität Konstanz, Universitätsstr. 10, 78464 Konstanz, Germany. Email address: franz.gmeineder@uni-konstanz...

work page doi:10.1007/s00211-023-01348-w

[3] [849]

On some techniques for approximating boundary conditions in the finite element method

doi: 10.1093/imanum/dry097. 58 REFERENCES [Ste95] R. Stenberg. “On some techniques for approximating boundary conditions in the finite element method”. Vol. 63. 1-3. International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994). 1995, pp. 139–148.doi: 10.1016/ 0377-0427(95)00057-7. [Sto45] G. G. Stokes. “On the th...

arXiv 1973

[4] [2023]

Wallslipofmoltenpolymers

doi: 10.25561/104839. [Hat12] S.G.Hatzikiriakos.“Wallslipofmoltenpolymers”. Progress in Polymer Science37.4(2012). Topical Issue on Polymer Physics, pp. 624–643. doi: https : / / doi . org / 10 . 1016 / j . progpolymsci.2011.09.004. [HCJ21] W. Han, K. Czuprynski, and F. Jing. “Mixed finite element method for a hemivariational inequality of stationary Navi...

work page arXiv 2012

[5] [2212]

A fully asynchronous mul- tifrontal solver using distributed dynamic scheduling

doi: 10.1142/S0218202521500470. [ADLK01] P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, and J. Koster. “A fully asynchronous mul- tifrontal solver using distributed dynamic scheduling”.SIAM J. Matrix Anal. Appl.23.1 (2001), pp. 15–41.doi: 10.1137/S0895479899358194. [AZ24] R. Altmann and C. Zimmer. “A second-order bulk-surface splitting for parabolic proble...

work page arXiv 2001

[6] [5386]

Monotone operators in divergence form with x- dependent multivalued graphs

doi: 10.3934/dcds.2016036. [FMT04] G. Francfort, F. Murat, and L. Tatar. “Monotone operators in divergence form with x- dependent multivalued graphs”.Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)7-B.1 (Feb. 2004), pp. 23–59. [FS95] J. Freund and R. Stenberg. “On weakly imposed boundary conditions for second order problems”. Finite elements in fluids...

work page doi:10.3934/dcds.2016036 2012