A Quadratic $C^0$ Interior Penalty Method for the von K\'{a}rm\'{a}n Obstacle Problem

Sharat Gaddam

REVIEW 3 major objections 3 minor 45 references

A Quadratic $C^0$ Interior Penalty Method for the von K\'{a}rm\'{a}n Obstacle Problem

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A quadratic $C^0$ interior penalty method converges at order $O(h^\alpha)$ for the von Kármán obstacle problem.

desk verdict Solid, workmanlike extension of C0IP to the von Kármán obstacle problem; the error analysis is clean but conditional on an imported H^{2+α} regularity theorem that a referee should check. read the letter →

arxiv 2608.10507 v1 pith:EOLNBMVF submitted 2026-08-11 math.NA cs.NA

Sharat Gaddam This is my paper · ORCID

classification math.NAcs.NA MSC 65N3065N1265N1574K2065K1549J40

keywords vonKármánequationsdisplacementobstacleproblemC0interiorpenaltymethodfourth-ordervariationalinequalityapriorierrorestimateprimal–dualactivesetstrategycoincidence

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 analyzes a finite element method for a thin plate held above an obstacle, where the plate is governed by the nonlinear von Kármán equations and the contact region is unknown. The method uses continuous piecewise quadratic functions, enforces the obstacle only at triangle vertices, and adds edge terms to the von Kármán bracket so that the discrete trilinear form is bounded in the discrete energy norm. The main result is that the discrete problem has a solution, that the solution is unique when the data are small enough, and that the energy-norm error decays as $O(h^\alpha)$, where $\alpha\in(1/2,1]$ is the elliptic regularity index of the polygonal domain. This gives a convergence theory for a quadratic $C^0$ interior penalty discretization of the displacement obstacle problem for von Kármán plates.

What carries the argument

The central object is the discrete energy norm $\|\cdot\|_h$, assembled from the piecewise Hessian $L^2$ norm, edge averages of the Hessian, and the weighted jump of the normal derivative across edges. The modified trilinear form $\mathcal F_{IP}$ adds edge terms to the piecewise von Kármán bracket and is bounded in this norm; it is symmetric in its first two arguments, which is what licenses the structure of the discrete problem. Two operators carry the proof: the nodal interpolation $I_h$ onto $P_2$ elements, and the enrichment $E_h$ from $V_h$ into the Argyris space that preserves vertex values. Because the discrete admissible set enforces the obstacle only at vertices, preservation of vertex values is exactly what makes $E_h$ map admissible discrete functions into the continuous admissible set.

What would settle it

On an L-shaped domain with the obstacle of (6.5), run the method on a sequence of uniformly refined meshes and compute the empirical order in the discrete energy norm. Theorem 5.2 predicts a rate of at least $0.5445$; if the empirical rate falls below $0.5445$ on sufficiently fine meshes, outside the pre-asymptotic range shown in Table 3, the regularity hypothesis or the error analysis is wrong.

Watch

Extended reading notes

Core claim

The paper proves that the discrete problem (4.1) is well posed for the quadratic $C^0$ interior penalty method: the discrete energy attains a minimizer over the vertex-constrained set $\mathcal E_h$, and the pair $(u_h,\varphi_h)$ solving the discrete complementarity system is unique whenever the data satisfy $C_{dS}\mathcal R_d(f,\chi)<1/2$. Under the same smallness condition and a sufficiently large penalty parameter $\sigma$, Theorem 5.2 establishes the energy-norm error estimate $\|u-u_h\|_h+\|\varphi-\varphi_h\|_h\le C h^\alpha$, where $\alpha\in(1/2,1]$ is the index of elliptic regularity of the biharmonic operator on the polygonal domain. The argument runs through a Strang-type consistency estimate in which the constraint error is controlled by the vertex-constrained auxiliary problem and the nonconformity is controlled by the Argyris enrichment operator.

Load-bearing premise

All error estimates rest on the imported regularity theorem that the continuous solution $(u,\varphi)$ belongs to $H^{2+\alpha}(\Omega)\cap W^3_{\rm loc}(\Omega)\cap C^2(\Omega)$ on polygonal domains; if the solution were only $H^{2+s}$ with $s<\alpha$, the claimed $O(h^\alpha)$ rate would not follow.

Editorial extensions

If this is right

On a convex polygon, where $\alpha=1$, the method converges with order $O(h)$ in the discrete energy norm for both solution components; on an L-shaped domain the guaranteed rate drops to $h^{0.5445}$.
The vertex-constrained $P_2$ space is sufficient: the obstacle can be imposed only at vertices, and the resulting consistency error is absorbed at order $O(h)$ by comparing with the auxiliary obstacle problem.
The penalty parameter $\sigma$ trades accuracy: increasing it improves the observed order for the displacement while lowering the order for the Airy stress function, and values beyond $\sigma=20$ push the stress function below the guaranteed rate on the square.
When the smallness condition fails, the method may still terminate in practice, but the uniqueness guarantee is withdrawn; the iterative solver fails first on the finest meshes as the obstacle is scaled up.

Reading between the lines

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

The observed pre-asymptotic rates above $\alpha$ on the L-shaped domain suggest the regularity theorem may not be sharp for this obstacle; running the method adaptively, or on a domain with a stronger corner singularity, would reveal whether the true limiting exponent is $h^{0.5445}$ or higher.
The explicit mesh-size dependence of the discrete Sobolev and Friedrichs constants in Theorem 3.4 could serve as the starting point for a posteriori error indicators and adaptive mesh refinement, which the paper leaves open.
The threshold behavior in Table 5 suggests the discrete smallness condition is sufficient but not necessary; a numerical check of the linearized operator's spectrum could map the actual uniqueness region.
The analysis identifies the vertex constraint as the natural discrete obstacle condition for $P_2$ elements; changing to edge-midpoint or interior point constraints would break the simple active-set structure used in the solver.

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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

This is a competent, honest paper. It does what it says: a quadratic C0 interior penalty method for the displacement obstacle problem of the von Kármán plate, with P2 elements, vertex constraints, an edge-modified trilinear form, well-posedness under a smallness condition, an O(h^α) energy-norm error estimate, and careful numerical experiments on a square and an L-shaped domain. The genuinely new thing is the combination of two existing analyses—Brenner et al. for C0IP for unconstrained von Kármán, and Brenner et al. for C0IP for the biharmonic obstacle problem—plus explicit mesh-dependent discrete Sobolev and Friedrichs constants. The proof is long but follows the established template, and the enrichment and interpolation arguments seem carefully executed. The experiments are notably honest: on the L-shape the observed rates are around 1.37, far above the guaranteed 0.54, and the authors explicitly label this pre-asymptotic rather than overclaiming. The penalty-parameter dependence in Table 4 is a nice touch, and the choice σ=10 is justified in a sensible worst-case way.

The central soft spot is the imported regularity Theorem 2.3 from Carstensen et al. (2021), which states that solutions of the continuous problem belong to H^{2+α}(Ω) with the same α as the biharmonic regularity index. That theorem is doing the heavy lifting, and the paper gives neither a proof nor a sketch. For the clamped plate obstacle problem, free-boundary regularity is a known subtlety: Caffarelli–Friedman deliver C^{1,1}, and the Hessian can jump, so H^{2+s} only for s<1/2 in rough cases. If the same limitation applies to the von Kármán obstacle problem, then the interpolation and enrichment bounds that drive Theorem 5.2 would only give O(h^s). The paper does not treat the free boundary directly, so the O(h^α) rate stands or falls with that imported regularity. I cannot tell from the present manuscript whether the Carstensen et al. theorem is true as stated, but this is the first thing I would ask a referee to verify.

There are also genuine but minor presentation errors: the proof of Theorem 5.2 refers to Theorem 3.5 and Theorem 5.1, where only Lemma 3.5 and Lemma 5.1 exist. That is confusing but harmless. No code is included; the tables are reasonably reproducible, but code would help.

This paper is for numerical analysts working on nonconforming discretizations of fourth-order obstacle problems. It is not a transformative contribution, but it is a useful and careful extension. I would send it to peer review, and I would likely accept after minor revision, with the regularity check as the main substantive request.

Referee Report

3 major / 3 minor

Summary. The paper proposes a quadratic C0 interior penalty method for the von Kármán displacement obstacle problem. The discrete space is the Lagrange P2 space with the obstacle constraint imposed at vertices, and the trilinear von Kármán bracket is modified by edge terms so that it is bounded in the discrete energy norm. The paper establishes existence of a discrete solution and its uniqueness under a smallness condition on the data, quantifies the discrete Sobolev and Friedrichs constants with explicit mesh-size dependence, and proves an a priori error estimate of order O(h^α) in the discrete energy norm, where α∈(1/2,1] is the biharmonic elliptic regularity index of the polygonal domain. Numerical experiments on a square and on an L-shaped domain, including two different types of coincidence sets, are reported, together with a study of the influence of the penalty parameter and of failure of the smallness condition.

Significance. If correct, the paper fills a genuine gap: it provides the first convergence analysis of a quadratic C0 interior penalty method for the von Kármán displacement obstacle problem. The technical apparatus is substantial and mostly carefully constructed: the edge-modified trilinear form, the interpolation/enrichment estimates, the explicit discrete embedding constants, and the treatment of the vertex-constrained obstacle all fit together coherently. The numerical section is honest about pre-asymptotic behavior, the dependence on the penalty parameter, and the failure of the iterative solver under large data. The principal risk is the reliance on the imported regularity theorem for the continuous problem; the convergence rate is no better than the regularity assumption that feeds into it.

major comments (3)

[Theorem 2.3 / Lemma 3.1 / Theorem 5.2] The rate O(h^α) with α>1/2 is driven entirely by Theorem 2.3, imported from (Carstensen et al., 2021) without proof, which asserts that every solution of (1.2) belongs to H^{2+α}(Ω)∩W^3_loc(Ω)∩C^2(Ω) on a polygonal domain. Lemma 3.1(i) produces the O(h^α) interpolation estimate only for H^{2+α} functions; Lemma 3.5 and the bounds in (5.13) require H^{2+α}⊂W^{2,4}; and the constants in Theorem 5.2 depend on ‖u‖_{2+α} and ‖φ‖_{2+α}. The proof of Theorem 5.2 contains no free-boundary argument, so the only source of the needed regularity is the imported theorem. This is not circular, because the prior result is published, but it is a load-bearing correctness risk: for fourth-order obstacle problems, C^{1,1} regularity at the free boundary would typically place u in H^{2+s} only for s<1/2, in which case the claimed rate does not follow from the argument given. The authors should either prove or explicitly assume the H^{2+α} regularity, or state the main theorem conditionally on it.
[Theorem 5.2] The hypotheses of Theorem 5.2 do not match those of Theorem 2.3. Theorem 5.2 assumes only χ∈C^2(Ω), whereas Theorem 2.3 requires χ∈H^2(Ω)∩H^3_loc(Ω)∩C^2(Ω), and the proof of Theorem 5.2 invokes Theorem 2.3 at the beginning of Section 5.1. The statement of Theorem 5.2 should include the stronger regularity assumptions on the obstacle, or a separate argument showing that C^2 regularity is sufficient for the invoked regularity conclusion.
[Section 5.2, after Eq. (5.14)] The proof defines r(ε)= (1/(2ε))C_dS^2‖u_h‖_h^2 + 2C_dS‖φ_h‖_h and states that 'Theorem 4.1 gives r(1)<1/2'. From the displayed uniqueness condition in Theorem 4.1(b), namely C_dS^2/4 ‖u_h‖_h^2 + C_dS‖φ_h‖_h < 1/2, one obtains r(1)=2 times the left-hand side, hence only r(1)<1. The assertion r(1)<1/2 is therefore not justified. The argument can be repaired by choosing ε0∈(0,1) with r(ε0)<1, but as written this step of the proof is incorrect.

minor comments (3)

[Throughout Sections 3 and 5] Several results are cited under the wrong designation: Lemmas 3.1, 3.2, 3.5, 3.6, 3.7 and 5.1 are repeatedly called Theorems 3.1, 3.2, 3.5, 3.6, 3.7 and 5.1, for example in Section 3.2 and in Steps 2 and 3 of the proof of Theorem 5.2. These cross-references should be corrected globally.
[Theorem 3.4 proof] The proof of Theorem 3.4 refers to 'Theorem 3.1' and 'Theorem 3.2'; these should be Lemma 3.1 and Lemma 3.2. Without the correction, the reader cannot locate the cited estimates.
[Lemma 5.1] Lemma 5.1 is imported from (Brenner et al., 2012b) without a proof or a precise statement of the assumptions on the obstacle that the source requires. Since the lemma supplies the O(h) and O(h^2) auxiliary-problem rates used in (5.10) and (5.11), the dependency should be stated as explicitly as the dependency on Theorem 2.3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the error estimate is a standard a priori analysis whose only imported ingredient is an externally published regularity theorem, not a restatement of the target result.

full rationale

The paper's derivation chain is: import existence, uniqueness and H^{2+alpha} regularity for the continuous problem from (Carstensen et al., 2021) as Theorems 2.2 and 2.3; construct the C0IP discrete problem (4.1); prove discrete well-posedness in Theorem 4.1; and then prove the energy-norm estimate in Theorem 5.2 through interpolation, enrichment, auxiliary obstacle problems and Young inequalities. None of the displayed estimates (5.3)-(5.16) defines the target error in terms of itself, and no fitted parameter is relabelled as a prediction. The penalty parameter sigma=10 is chosen in the experiments by a 'best worst case' criterion, but the theory only requires sigma sufficiently large, and the experimental rates are not used in the proof. The one load-bearing imported result is Theorem 2.3, taken from (Carstensen et al., 2021, Thm. 3.5), a peer-reviewed paper co-authored by the present author. Under the stated rules this is still independent support: that theorem does not assume the discrete error result, is externally published, and is falsifiable through regularity considerations. The paper explicitly says Section 2 gives no proofs, and Section 7 records open questions about the active-set/Newton solver and a posteriori analysis; these are limitations rather than circular steps. The numerical experiments compare discrete solutions against the finest-level discrete solution, which is a self-referential experimental practice, but it does not enter the derivation of Theorem 5.2. Therefore no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The theoretical result uses no physical free parameters beyond the penalty parameter. The paper imports the continuous regularity and uniqueness theorems from (Carstensen et al., 2021), a paper the author co-authored, which is legitimate prior support but is not independently machine-checked here.

free parameters (1)

penalty parameter σ = σ=10 in experiments (range 5-45 tested)
The discrete bilinear form E_IP includes the penalty term (σ/|e|) ∫ [∂v_h/∂n]^2. Theory requires σ sufficiently large, with σ=5 known to be coercive; the experiments scan σ and choose 10 because it maximizes the smallest ratio of observed order to guaranteed order across the six table columns. This is a hand-tuned method parameter, not a fitted physical constant.

assumptions (4)

domain assumption The continuous von Kármán obstacle problem (1.2) has at least one solution, a priori bounds, and uniqueness under C_S R(f,χ)<1/2, as stated in Theorem 2.2 from (Carstensen et al., 2021).
The discrete well-posedness proof mirrors the continuous one and inherits these existence and smallness conditions.
domain assumption The continuous solution (u,φ) on a polygonal domain lies in H^{2+α}(Ω) ∩ W^3_loc(Ω) ∩ C^2(Ω) for the biharmonic regularity index α, stated as Theorem 2.3 from (Carstensen et al., 2021, Thm 3.5).
Lemma 3.1's interpolation estimates and the main error theorem require this H^{2+α} regularity; without it the rate O(h^α) is not justified.
standard math Shift theorem for the biharmonic Dirichlet problem on polygonal domains with data in negative-order spaces (Bacuta et al., 2002, Thm 8; Blum et al., 1980, Thm 2).
Used in the proof of Theorem 3.4 to quantify the discrete Sobolev and Friedrichs constants with explicit powers of the mesh size.
domain assumption Shape-regular admissible triangulations with red refinement, so that P2 spaces are nested on successive meshes.
The interpolation and enrichment lemmas (Lemma 3.1 and Lemma 3.2) and the numerical setup in Section 6 assume these mesh properties.

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Pith. "Pith review of A Quadratic $C^0$ Interior Penalty Method for the von K\'{a}rm\'{a}n Obstacle Problem." pith.science (2026). https://pith.science/paper/EOLNBMVF

@misc{pith2026260810507,
  author       = {Pith},
  title        = {Pith review of: A Quadratic $C^0$ Interior Penalty Method for the von K\'arm\'an Obstacle Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOLNBMVF}},
  note         = {Machine review of arXiv:2608.10507}
}

abstract

This article proposes and analyses a quadratic $C^0$ interior penalty method for the displacement obstacle problem of the von K\'arm\'an plate. The discrete space consists of Lagrange $P_2$ finite elements and the obstacle constraint is imposed at the vertices. The trilinear form of the von K\'arm\'an bracket is modified by terms on the edges so that it is bounded in the discrete energy norm. The well-posedness of the discrete problem, namely the existence of a discrete solution and its uniqueness under a smallness condition on the data, is established. The Sobolev and Friedrichs constants of the discrete energy norm are quantified with an explicit dependence on the mesh size. The main result is an error estimate of order $\mathcal{O}(h^{\alpha})$ in the discrete energy norm, where $1/2<\alpha\le1$ is the index of elliptic regularity of the biharmonic operator on the polygonal domain. Numerical experiments on a square and on an L-shaped domain confirm the predicted rates. The coincidence set has positive measure in one example and empty interior in another. The experiments also show how the penalty parameter affects the rates and identify a threshold in the size of the obstacle beyond which the iterative solver fails on fine meshes.

Figures

Figures reproduced from arXiv: 2608.10507 by the authors.

**Figure 1.** Example 6.1 at = 10: the discrete displacement against the obstacle on the finest level, and the discrete coincidence sets on the two finest levels. Alt text: Three panels for the square with obstacle = 1 − 5|| 2 + || 4 . The upper panel is a surface plot of the discrete displacement lying above the obstacle surface. The lower two panels are scatter plots of the coincidence set at levels six and seven; both are fill… view at source ↗

**Figure 2.** Example 6.2 at = 10: the discrete displacement against the obstacle on the finest level, and the discrete coincidence sets on the two finest levels, which degenerate to a curve. Alt text: Three panels for the square with obstacle = 1 − 5|| 2 − || 4 . The upper panel is a surface plot of the discrete displacement lying above the obstacle surface. The lower two panels are scatter plots of the coincidence set at levels… view at source ↗

**Figure 3.** L-shaped domain at = 10: the discrete displacement against the obstacle on the finest level, and the discrete coincidence sets on the two finest levels. Alt text: Three panels for the L-shaped domain. The upper panel is a surface plot of the discrete displacement lying above the obstacle surface. The lower two panels show the coincidence set at levels five and six as a single compact blob in the left part of the dom… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Observed orders of convergence as functions of the [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

45 extracted references · 42 canonical work pages

[1]

2003 , publisher=

Sobolev spaces , author=. 2003 , publisher=

2003
[2]

Morley finite element method for the von K

Carstensen, Carsten and Gaddam, Sharat and Nataraj, Neela and Pani, Amiya K and Shylaja, Devika , journal=. Morley finite element method for the von K. 2021 , publisher=

work page 2021
[3]

SIAM journal on mathematical analysis , volume=

Stability in obstacle problems for the von Karman plate , author=. SIAM journal on mathematical analysis , volume=. 1992 , publisher=

work page 1992
[4]

Recent Progress in Computational and Applied PDEs , pages=

Shift theorems for the biharmonic Dirichlet problem , author=. Recent Progress in Computational and Applied PDEs , pages=. 2002 , publisher=

work page 2002
[5]

Morley Finite Element Method for the von K\'{a}rm\'{a}n Obstacle Problem

Morley Finite Element Method for the von K ' \ a \ rm ' \ a \ n Obstacle Problem , author=. arXiv preprint arXiv:2009.03205 , year=

work page Pith review arXiv 2009
[6]

Berger, Melvyn S and Fife, Paul C , journal=. Von. 1968 , publisher=

work page 1968
[7]

Mathematical Methods in the Applied Sciences , volume=

On the boundary value problem of the biharmonic operator on domains with angular corners , author=. Mathematical Methods in the Applied Sciences , volume=. 1980 , publisher=

work page 1980
[8]

Brenner, Susanne C and Neilan, Michael and Reiser, Armin and Sung, Li-Yeng , journal=. A. 2017 , publisher=

work page 2017

Show all 45 references

[9]

and Friedman, Avner , TITLE =

Caffarelli, Luis A. and Friedman, Avner , TITLE =. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , FJOURNAL =. 1979 , NUMBER =

1979
[10]

The primal-dual active set strategy as a semismooth

Hinterm\". The primal-dual active set strategy as a semismooth. SIAM J. Optim. , FJOURNAL =. 2002 , NUMBER =. doi:10.1137/S1052623401383558 , URL =

2002 doi
[11]

, author=

The mathematical theory of finite element methods. , author=. Texts in applied mathematics , year=
[12]

A quadratic

Brenner, Susanne C and Sung, Li-Yeng and Zhang, Hongchao and Zhang, Yi , journal=. A quadratic. 2012 , publisher=

2012
[13]

Journal of Computational and Applied Mathematics , volume=

A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates , author=. Journal of Computational and Applied Mathematics , volume=. 2013 , publisher=

2013
[14]

Mathematics of Computation , volume=

Finite element methods for the displacement obstacle problem of clamped plates , author=. Mathematics of Computation , volume=
[15]

Finite element approximations of the von

Brezzi, Franco , journal=. Finite element approximations of the von. 1978 , publisher=

1978
[16]

Numerische Mathematik , volume=

Error estimates for the finite element solution of variational inequalities , author=. Numerische Mathematik , volume=. 1977 , publisher=

1977
[17]

Annali della Scuola Normale Superiore di Pisa-Classe di Scienze , volume=

The obstacle problem for the biharmonic operator , author=. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze , volume=
[18]

Computers & Mathematics with Applications , volume=

A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes , author=. Computers & Mathematics with Applications , volume=. 2014 , publisher=

2014
[19]

arXiv preprint arXiv:1708.07627 , year=

Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity , author=. arXiv preprint arXiv:1708.07627 , year=

arXiv
[20]

A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von

Carstensen, Carsten and Mallik, Gouranga and Nataraj, Neela , journal=. A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von. 2018 , publisher=

2018
[21]

Adaptive Morley FEM for the von

Carstensen, Carsten and Nataraj, Neela , journal=. Adaptive Morley FEM for the von
[22]

arXiv preprint arXiv:1808.03535 , year=

How to prove the discrete reliability for nonconforming finite element methods , author=. arXiv preprint arXiv:1808.03535 , year=

arXiv
[23]

1997 , publisher=

Mathematical Elasticity: Volume II: Theory of Plates , author=. 1997 , publisher=

1997
[24]

1978 , publisher=

The finite element method for elliptic problems , author=. 1978 , publisher=

1978
[25]

Mathematics of Computation , volume=

Error estimates for the approximation of a class of variational inequalities , author=. Mathematics of Computation , volume=
[26]

Zum differenzierbarkeitsproblem bei variationsungleichungen h

Frehse, Jens , booktitle=. Zum differenzierbarkeitsproblem bei variationsungleichungen h. 1971 , organization=

1971
[27]

Manuscripta Mathematica , volume=

On the regularity of the solution of the biharmonic variational inequality , author=. Manuscripta Mathematica , volume=. 1973 , publisher=

1973
[28]

IMA Journal of Numerical Analysis , volume=

Morley finite element method for the eigenvalues of the biharmonic operator , author=. IMA Journal of Numerical Analysis , volume=. 2014 , publisher=

2014
[29]

2008 , publisher=

Lectures on numerical methods for non-linear variational problems , author=. 2008 , publisher=

2008
[30]

2009 , publisher=

Theoretical Numerical Analysis: A Functional Analysis Framework , author=. 2009 , publisher=

2009
[31]

SIAM journal on numerical analysis , volume=

Adaptive multilevel methods for obstacle problems , author=. SIAM journal on numerical analysis , volume=. 1994 , publisher=

1994
[32]

1980 , publisher=

An introduction to variational inequalities and their applications , author=. 1980 , publisher=

1980
[33]

An existence theorem for the von

Knightly, George H , journal=. An existence theorem for the von. 1967 , publisher=

1967
[34]

Conforming finite element methods for the von K

Mallik, Gouranga and Nataraj, Neela , journal=. Conforming finite element methods for the von K. 2016 , publisher=

2016
[35]

A nonconforming finite element approximation for the von

Mallik, Gouranga and Nataraj, Neela , journal=. A nonconforming finite element approximation for the von. 2016 , publisher=

2016
[36]

A mixed finite element method for the solution of the von

Miyoshi, Tetsuhiko , journal=. A mixed finite element method for the solution of the von. 1976 , publisher=

1976
[37]

A unilateral contact model with buckling in von

Muradova, Aliki D and Stavroulakis, Georgios E , journal=. A unilateral contact model with buckling in von. 2007 , publisher=

2007
[38]

Analysis of certain unilateral problems in von

Ohtake, K and Oden, J Tinsley and Kikuchi, Noboru , journal=. Analysis of certain unilateral problems in von. 1980 , publisher=

1980
[39]

Analysis of certain unilateral problems in von K

Ohtake, Kunihiko and Oden, J Tinsley and Kikuchi, Noboru , journal=. Analysis of certain unilateral problems in von K. 1980 , publisher=

1980
[40]

Hybrid finite element methods for the von

Quarteroni, Alfio , journal=. Hybrid finite element methods for the von. 1979 , publisher=

1979
[41]

On the numerical analysis of the von

Reinhart, Laure , journal=. On the numerical analysis of the von. 1982 , publisher=

1982
[42]

2008 , publisher=

Numerical solution of variational inequalities by adaptive finite elements , author=. 2008 , publisher=

2008
[43]

Discontinuous

Wang, Fei and Han, Weimin and Cheng, Xiao-liang , journal=. Discontinuous. 2010 , publisher=

2010
[44]

Numerische Mathematik , volume=

On the quadratic finite element approximation to the obstacle problem , author=. Numerische Mathematik , volume=. 2002 , publisher=

2002
[45]

Obstacle problem for von

Yau, Shing-Tung and Gao, Yang , journal=. Obstacle problem for von. 1992 , publisher=

1992

Pith tools

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

[1] [1]

2003 , publisher=

Sobolev spaces , author=. 2003 , publisher=

2003

[2] [2]

Morley finite element method for the von K

Carstensen, Carsten and Gaddam, Sharat and Nataraj, Neela and Pani, Amiya K and Shylaja, Devika , journal=. Morley finite element method for the von K. 2021 , publisher=

work page 2021

[3] [3]

SIAM journal on mathematical analysis , volume=

Stability in obstacle problems for the von Karman plate , author=. SIAM journal on mathematical analysis , volume=. 1992 , publisher=

work page 1992

[4] [4]

Recent Progress in Computational and Applied PDEs , pages=

Shift theorems for the biharmonic Dirichlet problem , author=. Recent Progress in Computational and Applied PDEs , pages=. 2002 , publisher=

work page 2002

[5] [5]

Morley Finite Element Method for the von K\'{a}rm\'{a}n Obstacle Problem

Morley Finite Element Method for the von K ' \ a \ rm ' \ a \ n Obstacle Problem , author=. arXiv preprint arXiv:2009.03205 , year=

work page Pith review arXiv 2009

[6] [6]

Berger, Melvyn S and Fife, Paul C , journal=. Von. 1968 , publisher=

work page 1968

[7] [7]

Mathematical Methods in the Applied Sciences , volume=

On the boundary value problem of the biharmonic operator on domains with angular corners , author=. Mathematical Methods in the Applied Sciences , volume=. 1980 , publisher=

work page 1980

[8] [8]

Brenner, Susanne C and Neilan, Michael and Reiser, Armin and Sung, Li-Yeng , journal=. A. 2017 , publisher=

work page 2017