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arxiv: 2606.06172 · v1 · pith:7RSJLK2Wnew · submitted 2026-06-04 · 🧮 math.AP

Small deformations of a near cylindrical tube for the Canham-Helfrich Energy with applications to biological membranes

Charles M. Elliott , Carsten Gr\"aser , Philip J. Herbert This is my paper

Pith reviewed 2026-06-28 00:32 UTC · model grok-4.3

classification 🧮 math.AP
keywords Canham-Helfrich energysmall deformationsbiological membranesquadratic approximationEuler-Lagrange equationswell-posednesstube-like surfacesarea constraint
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The pith

A quadratic energy approximates the Canham-Helfrich energy for small deformations of near-cylindrical tubes modeled as graphs over a fixed reference surface.

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

The paper develops a quadratic functional that approximates the Canham-Helfrich energy for tube-like surfaces that undergo small deformations while respecting clamped boundaries and an area constraint. Because the deformations are small, the surface is represented as a graph over an undeformed cylindrical reference, which simplifies the energy to a quadratic form in the graph function. From this reduced energy the authors derive a Lagrangian and the associated Euler-Lagrange equations, then prove that these equations are well-posed in suitable function spaces. The construction is motivated by biological membranes deformed by localized forces or curvature changes. Numerical examples illustrate how the model can be solved in practice.

Core claim

For a tube-like surface with clamped boundary and area constraint, the Canham-Helfrich energy admits a quadratic approximation when the surface is expressed as a graph over a fixed undeformed cylinder; the resulting Euler-Lagrange equations are well-posed in appropriate spaces.

What carries the argument

The quadratic energy functional obtained by restricting the Canham-Helfrich energy to graph representations of small deformations around the cylindrical tube.

Load-bearing premise

The deformations remain small enough that the surface can always be written as a graph over the fixed undeformed reference surface.

What would settle it

Numerical computation of the full Canham-Helfrich energy on a sequence of successively smaller graph deformations whose quadratic approximation error fails to approach zero at the expected rate.

Figures

Figures reproduced from arXiv: 2606.06172 by Carsten Gr\"aser, Charles M. Elliott, Philip J. Herbert.

Figure 1
Figure 1. Figure 1: A plot of the error from the experiment in Section 6.2. We see that one appears to obtain the convergence as in [LL17], despite low regularity data [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An image showing Γ0(ρu), where u solves the point forcing problem outlined in Section 6.3.1. For the purposes of visualisation, we set ρ = 20. Here, the colour corresponds to the value of u, with blue being low values and red being high values. 6.4. Line forcing. Here we set L = 6 and R = 1, and consider γ = Γ0 ∩ {x ∈ R 3 : (x2 − R) 2 + (x 2 3 + x 2 1 ) 2 − x 2 3 − 10−6 = 0} and f to be determined by ⟨f, v… view at source ↗
Figure 3
Figure 3. Figure 3: An image showing Γ0(ρu), where u solves the point con￾straint problem outlined in Section 6.3.2. For the purposes of visuali￾sation, we set ρ = 0.2. Here, the colour corresponds to the value of u, with blue being low values and red being high values [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An image showing Γ0(ρu), where u solves the line forcing problem outlined in Section 6.4. For the purposes of visualisation, we set ρ = 0.2. Here, the colour corresponds to the value of u, with blue being low values and red being high values. 6.5. Phase field. Here we consider that c0(ϕ) = 20ϕ for the unknown phase function ϕ. The potential W is given by the obstacle potential, (27) W(s) = ( 1 − s 2 |s| ≤ … view at source ↗
Figure 5
Figure 5. Figure 5: An sequence of images showing Γ0(ρu), where u solves the phase field forcing problem outlined in Section 6.5. From top left to bottom right we show the resulting image with k = 2, 4, 6, 8. As the number of interfacial regions increases, the energy increases. For the purposes of visualisation, we set ρ = 0.2. Here, unlike the other experiments, the colour corresponds to the value of φ, with blue correspondi… view at source ↗
read the original abstract

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops a quadratic energy approximating the Canham-Helfrich functional for small deformations of a near-cylindrical tube-like surface subject to clamped boundary conditions and an area constraint. The deformed surface is represented as a graph over a fixed reference surface; a Lagrangian is formed, the Euler-Lagrange equations are derived, well-posedness of the resulting linear system is proved in suitable Sobolev spaces, and illustrative numerical examples are presented.

Significance. If the quadratic approximation and the well-posedness result are valid, the reduced model supplies a tractable framework for analyzing localized deformations of biological membranes driven by point forces or spontaneous-curvature inhomogeneities, complementing existing variational approaches in membrane biophysics.

minor comments (3)
  1. The transition from the full Canham-Helfrich integrand to the quadratic form (presumably obtained by second variation) should be written out explicitly, including the precise linearization of the mean-curvature and Gaussian-curvature terms under the graph representation.
  2. In the well-posedness argument, the coercivity estimate for the bilinear form on the subspace orthogonal to constants (enforcing the area constraint) relies on the reference surface being exactly cylindrical; the paper should quantify how small deviations from the cylinder affect the constants in the estimate.
  3. The numerical examples would be strengthened by a brief description of the finite-element discretization, the choice of mesh, and a convergence check against the analytic coercivity constant.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments appear in the provided report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by parametrizing small deformations as graphs over a fixed cylindrical reference surface (standard first-order normal displacement), expanding the Canham-Helfrich integrand to quadratic order in the graph function, enforcing the area constraint via a Lagrange multiplier whose linearization produces an integral condition, and proving well-posedness of the resulting linear elliptic system on H² via coercivity and standard estimates under clamped boundary conditions. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain; the quadratic energy is the explicit second variation and the well-posedness argument relies on classical elliptic theory independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, axioms, or invented entities are stated. The graph-over-fixed-surface assumption is the main structural premise.

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Works this paper leans on

125 extracted references · 17 canonical work pages

  1. [1]

    2015 , publisher=

    The paraview guide: a parallel visualization application , author=. 2015 , publisher=

    2015

  2. [2]

    doi:10.5281/zenodo.3706994 , url =

    Dedner, Andreas and Kloefkorn, Robert and Nolte, Martin , title =. doi:10.5281/zenodo.3706994 , url =

    work page doi:10.5281/zenodo.3706994

  3. [3]

    URLhttp://link

    Sander, Oliver , year =. DUNE — The Distributed and Unified Numerics Environment , ISBN =. doi:10.1007/978-3-030-59702-3 , journal =

    work page doi:10.1007/978-3-030-59702-3

  4. [4]

    arXiv preprint arXiv2502.21137 , title =

    Meiners, Alexander and Uecker, Hannes , date-added =. arXiv preprint arXiv2502.21137 , title =

    arXiv

  5. [5]

    Brenner, Susanne C and Sung, Li-Yeng , journal =

  6. [6]

    Curvature instability in membranes , volume =

    Leibler, S , journal =. Curvature instability in membranes , volume =

  7. [7]

    Existence of varifold minimizers for the multiphase

    Brazda, Katharina and Lussardi, Luca and Stefanelli, Ulisse , journal =. Existence of varifold minimizers for the multiphase

  8. [8]

    Modeling and computation of two phase geometric biomembranes using surface finite elements , volume =

    Elliott, Charles M and Stinner, Bj. Modeling and computation of two phase geometric biomembranes using surface finite elements , volume =. Journal of Computational Physics , number =

  9. [9]

    Computation of two-phase biomembranes with phase dependentmaterial parameters using surface finite elements , volume =

    Elliott, Charles M and Stinner, Bj. Computation of two-phase biomembranes with phase dependentmaterial parameters using surface finite elements , volume =. Communications in Computational Physics , number =

  10. [10]

    Gradient flow dynamics of two-phase biomembranes: sharp interface variational formulation and finite element approximation , volume =

    Barrett, John W and Garcke, Harald and N. Gradient flow dynamics of two-phase biomembranes: sharp interface variational formulation and finite element approximation , volume =. The SMAI Journal of computational mathematics , pages =

  11. [11]

    Phase field models for small deformations of biomembranes arising as

    Hatcher, Luke , school =. Phase field models for small deformations of biomembranes arising as

  12. [12]

    Blowey, James F and Elliott, Charles M , journal =. The

  13. [13]

    Study of phase-separation dynamics by use of cell dynamical systems

    Oono, Yoshitsugu and Puri, Sanjay , journal =. Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling , volume =

  14. [14]

    Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt , volume =

    Kuzmin, Peter I and Akimov, Sergey A and Chizmadzhev, Yuri A and Zimmerberg, Joshua and Cohen, Fredric S , journal =. Line tension and interaction energies of membrane rafts calculated from lipid splay and tilt , volume =

  15. [15]

    A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains , volume =

    Elliott, Charles M and Ranner, Thomas , journal =. A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains , volume =

  16. [16]

    Primal-dual active set methods for

    Blank, Luise and Garcke, Harald and Sarbu, Lavinia and Styles, Vanessa , journal =. Primal-dual active set methods for

  17. [17]

    An extension of the projected gradient method to a

    Blank, Luise and Rupprecht, Christoph , journal =. An extension of the projected gradient method to a

  18. [18]

    Elliott, C. M. and Hatcher, L. , doi =. Domain formation via phase separation for spherical biomembranes with small deformations , volume =. European Journal of Applied Mathematics , number =. 2021 , bdsk-url-1 =

    2021

  19. [19]

    On the sharp interface limit of a phase field model for near spherical two phase biomembranes , volume =

    Elliott, Charles M and Hatcher, Luke and Stinner, Bj. On the sharp interface limit of a phase field model for near spherical two phase biomembranes , volume =. Interfaces and Free Boundaries , number =

  20. [20]

    Gov , doi =

    Roie Shlomovitz and Nir S. Gov , doi =. Physical Model of Contractile Ring Initiation in Dividing Cells , url =. Biophysical Journal , number =. 2008 , bdsk-url-1 =

    2008

  21. [21]

    Gov, N. S. , doi =. Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces , url =. 2018 , bdsk-url-1 =. https://royalsocietypublishing.org/doi/pdf/10.1098/rstb.2017.0115 , journal =

    work page doi:10.1098/rstb.2017.0115 2018

  22. [22]

    Boundary value problems for a special

    Deckelnick, Klaus and Doemeland, Marco and Grunau, Hans-Christoph , journal =. Boundary value problems for a special

  23. [23]

    The Dune framework: Basic concepts and recent developments , url =

    Peter Bastian and Markus Blatt and Andreas Dedner and Nils-Arne Dreier and Christian Engwer and Ren\'e Fritze and Carsten Gr\"aser and Christoph Gr. The Dune framework: Basic concepts and recent developments , url =. Computers. 2021 , bdsk-url-1 =. doi:https://doi.org/10.1016/j.camwa.2020.06.007 , issn =

    work page doi:10.1016/j.camwa.2020.06.007 2021

  24. [24]

    Hitchhiker's guide to the fractional

    Di Nezza, Eleonora and Palatucci, Giampiero and Valdinoci, Enrico , journal =. Hitchhiker's guide to the fractional

  25. [25]

    Bending stiffness of lipid bilayers

    Fischer, Thomas M , journal =. Bending stiffness of lipid bilayers. III. Gaussian curvature , volume =

  26. [26]

    Observations on a submicroscopic basophilic component of cytoplasm , volume =

    Porter, Keith R , journal =. Observations on a submicroscopic basophilic component of cytoplasm , volume =

  27. [27]

    arXiv , author =:1810.05862 , primaryclass =

    Shear-driven instabilities of membrane tubes and dynamin-induced scission , year =. arXiv , author =:1810.05862 , primaryclass =

    arXiv

  28. [28]

    and Elliott, Charles M

    Barrett, John W. and Elliott, Charles M. , journal =. Finite element approximation of the Dirichlet problem using the boundary penalty method , volume =

  29. [29]

    Stream function formulation of surface Stokes equations , volume =

    Reusken, Arnold , journal =. Stream function formulation of surface Stokes equations , volume =

  30. [30]

    Polyharmonic boundary value problems: positivity preserving and nonlinear higher order elliptic equations in bounded domains , year =

    Gazzola, Filippo and Grunau, Hans-Christoph and Sweers, Guido , publisher =. Polyharmonic boundary value problems: positivity preserving and nonlinear higher order elliptic equations in bounded domains , year =

  31. [31]

    Differential geometry of curves and surfaces: revised and updated second edition , year =

    Do Carmo, Manfredo P , publisher =. Differential geometry of curves and surfaces: revised and updated second edition , year =

  32. [32]

    Free energy computations: A mathematical perspective , year =

    Stoltz, Gabriel and Rousset, Mathias and others , publisher =. Free energy computations: A mathematical perspective , year =

  33. [33]

    Formation and Stability of Lipid Membrane Nanotubes , url =

    Bahrami, Amir Houshang and Hummer, Gerhard , doi =. Formation and Stability of Lipid Membrane Nanotubes , url =. 2017 , bdsk-url-1 =. https://doi.org/10.1021/acsnano.7b05542 , journal =

    work page doi:10.1021/acsnano.7b05542 2017

  34. [34]

    Weikl, Thomas R. , doi =. Membrane-Mediated Cooperativity of Proteins , url =. 2018 , bdsk-url-1 =. https://doi.org/10.1146/annurev-physchem-052516-050637 , journal =

    work page doi:10.1146/annurev-physchem-052516-050637 2018

  35. [35]

    Boundary value problems for variational integrals involving surface curvatures , volume =

    Nitsche, Johannes CC , journal =. Boundary value problems for variational integrals involving surface curvatures , volume =

  36. [36]

    Lipid membranes with an edge , volume =

    Capovilla, R and Guven, J and Santiago, JA , journal =. Lipid membranes with an edge , volume =

  37. [37]

    The physics of membrane tubes: soft templates for studying cellular membranes , url =

    Roux, Aur. The physics of membrane tubes: soft templates for studying cellular membranes , url =. Soft Matter , pages =. 2013 , bdsk-url-1 =. doi:10.1039/C3SM50514F , issue =

    work page doi:10.1039/c3sm50514f 2013

  38. [38]

    Nanotubular highways for intercellular organelle transport , volume =

    Rustom, Amin and Saffrich, Rainer and Markovic, Ivanka and Walther, Paul and Gerdes, Hans-Hermann , journal =. Nanotubular highways for intercellular organelle transport , volume =

  39. [39]

    Parametric approximation of

    Barrett, John W and Garcke, Harald and N. Parametric approximation of. SIAM Journal on Scientific Computing , number =

  40. [40]

    doi:10.1051/m2an/2021014 , journal =

    Stable approximations for axisymmetric. doi:10.1051/m2an/2021014 , journal =

    work page doi:10.1051/m2an/2021014

  41. [41]

    Minimising a relaxed

    Deckelnick, Klaus and Grunau, Hans-Christoph and R. Minimising a relaxed. Interfaces and Free Boundaries , number =

  42. [42]

    Classical solutions to the Dirichlet problem for

    Dall'Acqua, Anna and Deckelnick, Klaus and Grunau, Hans-Christoph , doi =. Classical solutions to the Dirichlet problem for. Advances in Calculus of Variations , month =. 2008 , bdsk-url-1 =

    2008

  43. [43]

    A Navier boundary value problem for

    Deckelnick, Klaus and Grunau, Hans-Christoph , journal =. A Navier boundary value problem for

  44. [44]

    Sch. The. Calculus of Variations and Partial Differential Equations , number =

  45. [45]

    Intercellular transfer mediated by tunneling nanotubes , url =

    Hans-Hermann Gerdes and Raquel Negr. Intercellular transfer mediated by tunneling nanotubes , url =. Current Opinion in Cell Biology , number =. 2008 , bdsk-url-1 =. doi:https://doi.org/10.1016/j.ceb.2008.03.005 , issn =

    work page doi:10.1016/j.ceb.2008.03.005 2008

  46. [46]

    Multifaceted roles of tunneling nanotubes in intercellular communication , volume =

    Marzo, Ludovica and Gousset, Karine and Zurzolo, Chiara , journal =. Multifaceted roles of tunneling nanotubes in intercellular communication , volume =

  47. [47]

    Formation and interaction of membrane tubes , volume =

    Der. Formation and interaction of membrane tubes , volume =. Physical review letters , number =

  48. [48]

    Multiple membrane tethers probed by atomic force microscopy , volume =

    Sun, Mingzhai and Graham, John S and Heged. Multiple membrane tethers probed by atomic force microscopy , volume =. biophysical journal , number =

  49. [49]

    Membrane tether formation from outer hair cells with optical tweezers , volume =

    Li, Zhiwei and Anvari, Bahman and Takashima, Masayoshi and Brecht, Peter and Torres, Jorge H and Brownell, William E , journal =. Membrane tether formation from outer hair cells with optical tweezers , volume =

  50. [50]

    Pulling long tubes from firmly adhered vesicles , url =

    D Cuvelier and N Chiaruttini and P Bassereau and P Nassoy , doi =. Pulling long tubes from firmly adhered vesicles , url =. Europhysics Letters (

  51. [51]

    Daniels, D. R. , doi =. Transport of solid bodies along tubular membrane tethers , url =. PLOS ONE , month =. 2019 , bdsk-url-1 =

    2019

  52. [52]

    arXiv , author =:1912.10317 , primaryclass =

    Domain formation via phase separation for spherical biomembranes with small deformations , year =. arXiv , author =:1912.10317 , primaryclass =

    arXiv 1912

  53. [53]

    and Rowlands, George and Sens, Pierre and Turner, Matthew S

    Al-Izzi, Sami C. and Rowlands, George and Sens, Pierre and Turner, Matthew S. , doi =. Hydro-osmotic Instabilities in Active Membrane Tubes , url =. Phys. Rev. Lett. , month =. 2018 , bdsk-url-1 =

    2018

  54. [54]

    The distributed and unified numerics environment, version 2.4 , volume =

    Blatt, Markus and Burchardt, Ansgar and Dedner, Andreas and Engwer, Christian and Fahlke, Jorrit and Flemisch, Bernd and Gersbacher, Christoph and Gr. The distributed and unified numerics environment, version 2.4 , volume =. Archive of Numerical Software , number =

  55. [55]

    arXiv , author =:1904.12074 , journal =

    Existence and Regularity of Spheres Minimising the. arXiv , author =:1904.12074 , journal =

    arXiv 1904

  56. [56]

    Larsson, Karl and Larson, Mats G. , doi =. A continuous/discontinuous. Math. Comp. , mrclass =. 2017 , bdsk-url-1 =

    2017

  57. [57]

    , isbn =

    Ciarlet, Philippe G. , isbn =. Linear and Nonlinear Functional Analysis with Applications , year =

  58. [58]

    and Herbert, Philip J

    Elliott, Charles M. and Herbert, Philip J. , doi =. Math. Comp. , title =. 2020 , bdsk-url-1 =

    2020

  59. [59]

    and Herbert, Philip J

    Elliott, Charles M. and Herbert, Philip J. , journal =. A formula for membrane mediated point particle interactions on near spherical biomembranes , volume =

  60. [60]

    Ordinary differential equations , url =

    Hartman, Philip , doi =. Ordinary differential equations , url =. 2002 , bdsk-url-1 =

    2002

  61. [61]

    Handbook of global optimization , volume =

    Horst, Reiner and Pardalos, Panos M , publisher =. Handbook of global optimization , volume =

  62. [62]

    Exponential convergence of Langevin distributions and their discrete approximations , volume =

    Roberts, Gareth O and Tweedie, Richard L and others , journal =. Exponential convergence of Langevin distributions and their discrete approximations , volume =

  63. [63]

    Variational principles and free-boundary problems , year =

    Friedman, Avner , isbn =. Variational principles and free-boundary problems , year =

  64. [64]

    Kent and Marijn G.J

    William Mike Henne and Helen M. Kent and Marijn G.J. Ford and Balachandra G. Hegde and Oliver Daumke and P. Jonathan G. Butler and Rohit Mittal and Ralf Langen and Philip R. Evans and Harvey T. McMahon , doi =. Structure and Analysis of FCHo2 F-BAR Domain: A Dimerizing and Membrane Recruitment Module that Effects Membrane Curvature , url =. Structure , nu...

    2007

  65. [65]

    FCHo proteins are nucleators of clathrin-mediated endocytosis , volume =

    Henne, William Mike and Boucrot, Emmanuel and Meinecke, Michael and Evergren, Emma and Vallis, Yvonne and Mittal, Rohit and McMahon, Harvey T , journal =. FCHo proteins are nucleators of clathrin-mediated endocytosis , volume =

  66. [66]

    and Hatcher, Luke and Herbert, Philip J

    Elliott, Charles M. and Hatcher, Luke and Herbert, Philip J. , booktitle =. Small deformations of spherical biomembranes , year =. 1911.02964 , publisher =

    arXiv 1911

  67. [67]

    Energetic variational approaches for incompressible fluid systems on an evolving surface , volume =

    Koba, Hajime and Liu, Chun and Giga, Yoshikazu , journal =. Energetic variational approaches for incompressible fluid systems on an evolving surface , volume =

  68. [68]

    Mixed finite element methods for linear elasticity with weakly imposed symmetry , volume =

    Arnold, Douglas and Falk, Richard and Winther, Ragnar , journal =. Mixed finite element methods for linear elasticity with weakly imposed symmetry , volume =

  69. [69]

    A domain mapping approach for elliptic equations posed on random bulk and surface domains , year =

    Church, Lewis and Djurdjevac, Ana and Elliott, Charles M , journal =. A domain mapping approach for elliptic equations posed on random bulk and surface domains , year =

  70. [70]

    Navier-Stokes equations: theory and numerical analysis , volume =

    Temam, Roger , publisher =. Navier-Stokes equations: theory and numerical analysis , volume =

  71. [71]

    and Fritz, Hans and Hobbs, Graham , fjournal =

    Elliott, Charles M. and Fritz, Hans and Hobbs, Graham , fjournal =. Small deformations of. Math. Models Methods Appl. Sci. , mrclass =

  72. [72]

    C. M. Elliott and H. Fritz and G. Hobbs , journal =. Second order splitting for a class of fourth order equations , volume =

  73. [73]

    and Walkington, Noel J

    Howell, Jason S. and Walkington, Noel J. , day =. Inf--sup conditions for twofold saddle point problems , volume =. Numerische Mathematik , month =. 2011 , bdsk-url-1 =. doi:10.1007/s00211-011-0372-5 , issn =

    work page doi:10.1007/s00211-011-0372-5 2011

  74. [74]

    Schatz , issn =

    Alfred H. Schatz , issn =. A Weak Discrete Maximum Principle and Stability of the Finite Element Method in L_ on Plane Polygonal Domains. I , url =. Mathematics of Computation , number =. 1980 , bdsk-url-1 =

    1980

  75. [75]

    Elliptic problems in nonsmooth domains , volume =

    Grisvard, Pierre , publisher =. Elliptic problems in nonsmooth domains , volume =

  76. [76]

    A note on a finite element method dealing with corner singularities , url =

    Kim, Seokchan and Woo, Gyungsoo and Park, Taehoon , day =. A note on a finite element method dealing with corner singularities , url =. Korean Journal of Computational. 2000 , bdsk-url-1 =. doi:10.1007/BF03012199 , issn =

    work page doi:10.1007/bf03012199 2000

  77. [77]

    and Pasciak, Joseph E

    Bramble, James H. and Pasciak, Joseph E. and Steinbach, Olaf , doi =. On the stability of the. Math. Comp. , mrclass =. 2002 , bdsk-url-1 =

    2002

  78. [78]

    and Scott, L

    Brenner, Susanne C. and Scott, L. Ridgway , doi =. The mathematical theory of finite element methods , url =. 2002 , bdsk-url-1 =

    2002

  79. [79]

    Casas, Eduardo , doi =. Numer. Math. , mrclass =. 1985 , bdsk-url-1 =

    1985

  80. [80]

    Riemannian geometry , year =

    Willmore, Thomas James , publisher =. Riemannian geometry , year =

Showing first 80 references.