Error analysis for a Finite Element Discretization of a corotational harmonic map heat flow problem

Arnold Reusken; Nam Anh Nguyen

arxiv: 2506.23748 · v2 · submitted 2025-06-30 · 🧮 math.NA · cs.NA

Error analysis for a Finite Element Discretization of a corotational harmonic map heat flow problem

Nam Anh Nguyen , Arnold Reusken This is my paper

Pith reviewed 2026-05-19 07:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords harmonic map heat flowcorotational casefinite element methodsemi-implicit Eulererror analysisoptimal order boundsdiscrete energy estimateconvexity property

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The pith

Finite element method with semi-implicit stepping yields optimal error bounds for corotational harmonic map heat flow under smooth solutions.

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

The paper performs an error analysis for a discretization of the corotational harmonic map heat flow that combines H1-conforming finite elements in space with semi-implicit Euler time stepping. It establishes optimal order error bounds in the regime where the exact solution remains smooth. The argument rests on a discrete energy estimate that replicates the continuous energy dissipation and a convexity property that guarantees stability while controlling the linearization error. This supplies rigorous justification for applying the scheme to geometric evolution problems where such flows model physical systems like liquid crystals.

Core claim

For the corotational harmonic map heat flow problem the H1-conforming finite element method in space combined with semi-implicit Euler time stepping produces optimal order discretization error bounds when the continuous solution stays smooth. The proof uses a discrete energy estimate that mimics the energy dissipation property of the continuous problem together with a convexity property that is essential for discrete stability and for controlling the linearization error.

What carries the argument

Discrete energy estimate combined with convexity property, which together replicate continuous dissipation, guarantee stability, and bound the linearization error.

If this is right

The numerical approximation converges at optimal rates in the appropriate norms.
Each time step reduces to the solution of a linear system.
Numerical experiments confirm the theoretical convergence rates.
The same stability ingredients apply to related parabolic geometric flows in the smooth regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same discrete energy and convexity arguments could extend to other harmonic map flows provided a comparable convexity structure holds.
In practice, monitoring the smoothness of computed solutions could serve as an indicator for when the optimal rates remain reliable.
The technique may inform error analyses for nearby problems such as mean curvature flow or liquid crystal director evolution.

Load-bearing premise

The exact solution of the continuous problem remains smooth throughout the time interval considered.

What would settle it

Run the scheme on a test case in which the continuous solution develops a singularity and check whether the observed convergence rate drops below the predicted optimal order.

Figures

Figures reproduced from arXiv: 2506.23748 by Arnold Reusken, Nam Anh Nguyen.

read the original abstract

We consider the harmonic map heat flow problem for a corotational case. For discretization of this problem we apply a $H^1$-conforming finite element method in space combined with a semi-implicit Euler time stepping. The semi-implicit Euler method results in a linear problem in each time step. We restrict to the regime of smooth solutions of the continuous problem and present an error analysis of this discretization method. This results in optimal order discretization error bounds. Key ingredients of the analysis are a discrete energy estimate, that mimics the energy dissipation of the continuous solution, and a convexity property that is essential for discrete stability and for control of the linearization error. We also present numerical results that validate the theoretical ones.

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.

Desk Editor's Note private letter to a colleague

Solid error analysis for FEM plus semi-implicit Euler on the corotational harmonic map heat flow, with optimal rates under smoothness, but the scope is narrow.

read the letter

The main thing to know is that this paper works out an error analysis for an H1-conforming finite element method in space paired with semi-implicit Euler in time for the corotational harmonic map heat flow. They obtain optimal-order bounds by establishing a discrete energy estimate that copies the continuous dissipation and by using a convexity property to get stability plus control on the linearization error. The analysis stays inside the regime of smooth solutions, and they add numerical tests that line up with the theory. Each time step stays linear, which is a practical feature. The combination of those two ingredients for this particular geometric flow looks like a genuine step beyond the references they cite, rather than a routine extension. The energy law and convexity argument are set up independently of the final error bound, so there is no obvious circularity. The numerics provide some external check on the rates. The clear limitation is the smoothness assumption. Harmonic map flows often form singularities, and once that happens the discrete energy control and convexity may stop delivering the stated rates. The paper does not explore what the scheme does or how the error behaves near singularities, which keeps the result inside a restricted setting. Within that setting the argument appears consistent and the ingredients are used in a direct way. This is a paper for people who work on numerical methods for nonlinear geometric PDEs or harmonic maps. A reader who needs a concrete, provably convergent discretization with error bounds for this exact problem, or who wants to see how discrete energy methods and convexity are applied to control linearization in such flows, will get something usable. It has enough technical substance and a focused claim that it should go to a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper analyzes a finite element discretization of the corotational harmonic map heat flow using an H¹-conforming method in space and semi-implicit Euler time stepping. Restricted to the regime of smooth solutions, it derives optimal-order discretization error bounds via a discrete energy estimate that mimics continuous dissipation and a convexity property for discrete stability and linearization control. Numerical results are included to validate the theoretical bounds.

Significance. If the analysis holds, the work provides a rigorous error estimate for a stable, linear-per-step discretization of a nonlinear geometric PDE. The explicit use of a discrete energy law and convexity property to control the linearization error is a standard and effective technique in this area; the restriction to smooth solutions is clearly stated and the numerical validation supports practical applicability within that regime.

minor comments (3)

The abstract states that optimal-order bounds are obtained but does not specify the precise rates (e.g., O(h + τ) or similar). Adding this detail would make the main result immediately clearer to readers.
In the numerical section, the choice of manufactured solutions or test cases and the precise norms used to compute the reported errors should be stated explicitly so that the observed rates can be directly compared to the theoretical predictions.
Notation for the discrete energy and the convexity constant should be introduced once and used consistently; occasional re-definition of symbols across sections can be avoided by a short notation table or paragraph.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The report accurately summarizes the main contributions, including the restriction to smooth solutions, the discrete energy estimate, and the role of the convexity property. Since the report lists no specific major comments, we provide no point-by-point responses below and will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent discrete energy law and convexity control

full rationale

The paper derives optimal-order error bounds for the FEM/semi-implicit Euler discretization under a smoothness assumption on the exact solution. The central steps are a discrete energy estimate that mimics the continuous dissipation law and a convexity property used for stability and linearization control. These are constructed directly from the discrete scheme and the problem structure rather than being fitted to the target error or defined in terms of the final bound. No self-citation is invoked as a load-bearing uniqueness theorem, no parameter is fitted on a subset and then renamed as a prediction, and the analysis remains self-contained within the stated regime. The numerical results serve only for validation, not as input to the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard finite-element approximation theory and on the assumption that the continuous solution remains smooth; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption The exact solution of the continuous problem remains sufficiently smooth for the error analysis to apply.
Explicitly stated in the abstract as the regime under consideration.

pith-pipeline@v0.9.0 · 5651 in / 1225 out tokens · 22362 ms · 2026-05-19T07:37:05.157841+00:00 · methodology

discussion (0)

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Key ingredients of the analysis are a discrete energy estimate, that mimics the energy dissipation of the continuous solution, and a convexity property that is essential for discrete stability and for control of the linearization error.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

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Akrivis, S

G. Akrivis, S. Bartels, and C. Palus, Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries , Math. Comp., (2024)

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Alouges and P

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S. Bartels, K. B ¨ohnlein, C. Palus, and O. Sander , Benchmarking numerical algorithms for harmonic maps into the sphere , arXiv:2209.13665, (2024)

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Bartels, B

S. Bartels, B. Kov ´acs, and Z. Wang , Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints , IMA J. Numer. Anal., 44 (2024), pp. 633–653

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S. Bartels, C. Lubich, and A. Prohl , Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers , Math. Comp., 78 (2009), pp. 1269–1292

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Bartels and A

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Bartels and A

S. Bartels and A. Prohl, Constraint preserving implicit finite element discretization of har- monic map flow into spheres , Math. Comp., 76 (2007), pp. 1847–1859

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M. Bertsch, R. Dal Passo, and R. van der Hout , Nonuniqueness for the heat flow of Harmonic Maps on the disk , Archive for Rational Mechanics and Analysis, 161 (2002), p. 93–112

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K.-C. Chang and W.-Y. Ding , A result on the global existence for heat flows of harmonic maps from D2 into S2 , in Nematics: Mathematical and Physical Aspects, J.-M. Coron, J.-M. Ghidaglia, and F. H´ elein, eds., Dordrecht, 1991, Springer Netherlands, pp. 37–47

work page 1991
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Chang, W

K.-C. Chang, W. Y. Ding, and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry, 36 (1992), pp. 507 – 515

work page 1992
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Cimr´ak, Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field , IMA J

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J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal 15 of Mathematics, 86 (1964), p. 109

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M. Guan, S. on, and T.-P. Tsai , Global existence and blow-up for harmonic map heat flow , Journal of Differential Equations, 246 (2009), pp. 1–20

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Gustafson, K

S. Gustafson, K. Nakanishi, and T.-P. Tsai, Asymptotic stability, concentration, and oscil- lation in harmonic map heat-flow, Landau-Lifshitz, and Schr¨ odinger maps on R2, Com- munications in Mathematical Physics, 300 (2010), p. 205–242

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Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

S. Gustafson and D. Roxanas , Global solutions for the critical, higher-degree corotational harmonic map heat flow to S2, arXiv:1711.06476, (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017
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R. S. Hamilton , Harmonic Maps of Manifolds with Boundary , Springer Berlin, Heidelberg, 1975

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Haynes, W

R. Haynes, W. Huang, and P. Zegeling , A numerical study of blowup in the harmonic map heat flow using the MMPDE moving mesh method , Numerical Mathematics: Theory, Methods and Applications, 6 (2013), pp. 364–383

work page 2013
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Heinze, K

S. Heinze, K. Bergmann, M. Menzel, and et al. , Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions , Nature Phys., 7 (2011), pp. 713–718

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work page 1989
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Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview , Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 369 (2011), pp

M. Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview , Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 369 (2011), pp. 1280–1300

work page 2011
[28]

Liu and Y.-Q

Y.-H. Liu and Y.-Q. Li, A mechanism to pin skyrmions in chiral magnets , J. Phys.: Condens. Matter, 25 (2013), p. 076005

work page 2013
[29]

Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions , Indiana University Mathematics Journal, 61 (2011), pp

C. Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions , Indiana University Mathematics Journal, 61 (2011), pp. 1175–1200

work page 2011
[30]

N. A. Nguyen, HMHF-solver, 2025. Zenodo. https://doi.org/10.5281/zenodo.15481333

work page doi:10.5281/zenodo.15481333 2025
[31]

N. A. Nguyen and A. Reusken , Discretization error analysis for a radially symmetric har- monic map heat flow problem , arXiv:2503.03525, (2025)

work page arXiv 2025
[32]

Prohl, Computational Micromagnetism, Vieweg+Teubner Verlag Wiesbaden, 2001

A. Prohl, Computational Micromagnetism, Vieweg+Teubner Verlag Wiesbaden, 2001

work page 2001
[33]

Rapha¨el and R

P. Rapha¨el and R. Schweyer , Stable blowup dynamics for the 1-corotational energy criti- cal harmonic heat flow , Communications on Pure and Applied Mathematics, 66 (2013), pp. 414–480

work page 2013
[34]

Roxanas, Long-time dynamics for the energy-critical Harmonic Map Heat Flow and Non- linear Heat Equation, PhD thesis, University of Britisch Columbia, Vancouver, May 2017

D. Roxanas, Long-time dynamics for the energy-critical Harmonic Map Heat Flow and Non- linear Heat Equation, PhD thesis, University of Britisch Columbia, Vancouver, May 2017

work page 2017
[35]

Struwe, On the evolution of harmonic mappings of Riemannian surfaces , Commentarii Mathematici Helvetici, 60 (1985), pp

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces , Commentarii Mathematici Helvetici, 60 (1985), pp. 558–581

work page 1985
[36]

Struwe, Variational Methods, Springer Berlin, Heidelberg, 2008

M. Struwe, Variational Methods, Springer Berlin, Heidelberg, 2008

work page 2008
[37]

Topping , Reverse bubbling and nonuniqueness in the harmonic map flow , International Mathematics Research Notices, 2002 (2002), pp

P. Topping , Reverse bubbling and nonuniqueness in the harmonic map flow , International Mathematics Research Notices, 2002 (2002), pp. 505–520

work page 2002
[38]

J. B. van den Berg, J. Hulshof, and J. R. King , Formal asymptotics of bubbling in the harmonic map heat flow , SIAM J. Appl. Math., 63 (2003), pp. 1682–1717

work page 2003
[39]

van der Hout, Flow alignment in nematic liquid crystals in flows with cylindrical symme- try, Differential and Integral Equations, 14 (2001), p

R. van der Hout, Flow alignment in nematic liquid crystals in flows with cylindrical symme- try, Differential and Integral Equations, 14 (2001), p. 189 – 211. 16

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[1] [1]

Akrivis, S

G. Akrivis, S. Bartels, and C. Palus, Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries , Math. Comp., (2024)

work page 2024

[2] [2]

Akrivis, S

G. Akrivis, S. Bartels, M. Ruggeri, and J. Wang , Projection-free approximation of flows of harmonic maps with quadratic constraint consistency and variable step sizes , arXiv:2505.05655, (2025)

work page arXiv 2025

[3] [3]

Akrivis, M

G. Akrivis, M. Feischl, B. Kov ´acs, and C. Lubich , Higher-order linearly implicit full dis- cretization of the Landau-Lifshitz-Gilbert equation, Math. Comp., 90 (2021), pp. 995–1038

work page 2021

[4] [4]

Alouges, A new finite element scheme for Landau-Lifshitz equations , Discrete and Contin- uous Dynamical Systems - S, 1 (2008), pp

F. Alouges, A new finite element scheme for Landau-Lifshitz equations , Discrete and Contin- uous Dynamical Systems - S, 1 (2008), pp. 187–196

work page 2008

[5] [5]

Alouges and P

F. Alouges and P. Jaisson, Convergence of a finite element discretization for Landau-Lifshitz equations in micromagnetism, Mathematical Models and Methods in Applied Sciences, 16 (2006), p. 299–316

work page 2006

[6] [6]

S. B. Angenent, J. Hulshof, and H. Matano, The radius of vanishing bubbles in equivariant harmonic map flow from D2 to S2, SIAM Journal on Mathematical Analysis, 41 (2009), pp. 1121–1137

work page 2009

[7] [7]

Bartels, K

S. Bartels, K. B ¨ohnlein, C. Palus, and O. Sander , Benchmarking numerical algorithms for harmonic maps into the sphere , arXiv:2209.13665, (2024)

work page arXiv 2024

[8] [8]

Bartels, B

S. Bartels, B. Kov ´acs, and Z. Wang , Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints , IMA J. Numer. Anal., 44 (2024), pp. 633–653

work page 2024

[9] [9]

Bartels, C

S. Bartels, C. Lubich, and A. Prohl , Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers , Math. Comp., 78 (2009), pp. 1269–1292

work page 2009

[10] [10]

Bartels and A

S. Bartels and A. Prohl, Convergence of an implicit finite element method for the Landau- Lifshitz-Gilbert equation, SIAM J. Numer. Anal., 44 (2006), pp. 1405–1419

work page 2006

[11] [11]

Bartels and A

S. Bartels and A. Prohl, Constraint preserving implicit finite element discretization of har- monic map flow into spheres , Math. Comp., 76 (2007), pp. 1847–1859

work page 2007

[12] [12]

Bertsch, R

M. Bertsch, R. Dal Passo, and R. van der Hout , Nonuniqueness for the heat flow of Harmonic Maps on the disk , Archive for Rational Mechanics and Analysis, 161 (2002), p. 93–112

work page 2002

[13] [13]

S. C. Brenner and L. R. Scott , The Mathematical Theory of Finite Element Methods , Springer New York, 2008

work page 2008

[14] [14]

Chang and W.-Y

K.-C. Chang and W.-Y. Ding , A result on the global existence for heat flows of harmonic maps from D2 into S2 , in Nematics: Mathematical and Physical Aspects, J.-M. Coron, J.-M. Ghidaglia, and F. H´ elein, eds., Dordrecht, 1991, Springer Netherlands, pp. 37–47

work page 1991

[15] [15]

Chang, W

K.-C. Chang, W. Y. Ding, and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry, 36 (1992), pp. 507 – 515

work page 1992

[16] [16]

Cimr´ak, Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field , IMA J

I. Cimr´ak, Error estimates for a semi-implicit numerical scheme solving the Landau–Lifshitz equation with an exchange field , IMA J. Numer. Anal., 25 (2005), pp. 611–634

work page 2005

[17] [17]

Eells and J

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal 15 of Mathematics, 86 (1964), p. 109

work page 1964

[18] [18]

A. R. Freire, Uniqueness for the harmonic map flow in two dimensions , Calculus of Variations and Partial Differential Equations, 3 (1995), pp. 95–105

work page 1995

[19] [19]

M. Guan, S. on, and T.-P. Tsai , Global existence and blow-up for harmonic map heat flow , Journal of Differential Equations, 246 (2009), pp. 1–20

work page 2009

[20] [20]

Gustafson, K

S. Gustafson, K. Nakanishi, and T.-P. Tsai, Asymptotic stability, concentration, and oscil- lation in harmonic map heat-flow, Landau-Lifshitz, and Schr¨ odinger maps on R2, Com- munications in Mathematical Physics, 300 (2010), p. 205–242

work page 2010

[21] [21]

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

S. Gustafson and D. Roxanas , Global solutions for the critical, higher-degree corotational harmonic map heat flow to S2, arXiv:1711.06476, (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017

[22] [22]

R. S. Hamilton , Harmonic Maps of Manifolds with Boundary , Springer Berlin, Heidelberg, 1975

work page 1975

[23] [23]

Haynes, W

R. Haynes, W. Huang, and P. Zegeling , A numerical study of blowup in the harmonic map heat flow using the MMPDE moving mesh method , Numerical Mathematics: Theory, Methods and Applications, 6 (2013), pp. 364–383

work page 2013

[24] [24]

Heinze, K

S. Heinze, K. Bergmann, M. Menzel, and et al. , Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions , Nature Phys., 7 (2011), pp. 713–718

work page 2011

[25] [25]

Hocquet, Finite-time singularity of the stochastic harmonic map flow , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 55 (2019), pp

A. Hocquet, Finite-time singularity of the stochastic harmonic map flow , Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 55 (2019), pp. 1011 – 1041

work page 2019

[26] [26]

Kung-Ching, Heat flow and boundary value problem for harmonic maps, Annales de l’I.H.P

C. Kung-Ching, Heat flow and boundary value problem for harmonic maps, Annales de l’I.H.P. Analyse non lin´ eaire, 6 (1989), pp. 363–395

work page 1989

[27] [27]

Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview , Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 369 (2011), pp

M. Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: an overview , Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 369 (2011), pp. 1280–1300

work page 2011

[28] [28]

Liu and Y.-Q

Y.-H. Liu and Y.-Q. Li, A mechanism to pin skyrmions in chiral magnets , J. Phys.: Condens. Matter, 25 (2013), p. 076005

work page 2013

[29] [29]

Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions , Indiana University Mathematics Journal, 61 (2011), pp

C. Melcher, Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions , Indiana University Mathematics Journal, 61 (2011), pp. 1175–1200

work page 2011

[30] [30]

N. A. Nguyen, HMHF-solver, 2025. Zenodo. https://doi.org/10.5281/zenodo.15481333

work page doi:10.5281/zenodo.15481333 2025

[31] [31]

N. A. Nguyen and A. Reusken , Discretization error analysis for a radially symmetric har- monic map heat flow problem , arXiv:2503.03525, (2025)

work page arXiv 2025

[32] [32]

Prohl, Computational Micromagnetism, Vieweg+Teubner Verlag Wiesbaden, 2001

A. Prohl, Computational Micromagnetism, Vieweg+Teubner Verlag Wiesbaden, 2001

work page 2001

[33] [33]

Rapha¨el and R

P. Rapha¨el and R. Schweyer , Stable blowup dynamics for the 1-corotational energy criti- cal harmonic heat flow , Communications on Pure and Applied Mathematics, 66 (2013), pp. 414–480

work page 2013

[34] [34]

Roxanas, Long-time dynamics for the energy-critical Harmonic Map Heat Flow and Non- linear Heat Equation, PhD thesis, University of Britisch Columbia, Vancouver, May 2017

D. Roxanas, Long-time dynamics for the energy-critical Harmonic Map Heat Flow and Non- linear Heat Equation, PhD thesis, University of Britisch Columbia, Vancouver, May 2017

work page 2017

[35] [35]

Struwe, On the evolution of harmonic mappings of Riemannian surfaces , Commentarii Mathematici Helvetici, 60 (1985), pp

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces , Commentarii Mathematici Helvetici, 60 (1985), pp. 558–581

work page 1985

[36] [36]

Struwe, Variational Methods, Springer Berlin, Heidelberg, 2008

M. Struwe, Variational Methods, Springer Berlin, Heidelberg, 2008

work page 2008

[37] [37]

Topping , Reverse bubbling and nonuniqueness in the harmonic map flow , International Mathematics Research Notices, 2002 (2002), pp

P. Topping , Reverse bubbling and nonuniqueness in the harmonic map flow , International Mathematics Research Notices, 2002 (2002), pp. 505–520

work page 2002

[38] [38]

J. B. van den Berg, J. Hulshof, and J. R. King , Formal asymptotics of bubbling in the harmonic map heat flow , SIAM J. Appl. Math., 63 (2003), pp. 1682–1717

work page 2003

[39] [39]

van der Hout, Flow alignment in nematic liquid crystals in flows with cylindrical symme- try, Differential and Integral Equations, 14 (2001), p

R. van der Hout, Flow alignment in nematic liquid crystals in flows with cylindrical symme- try, Differential and Integral Equations, 14 (2001), p. 189 – 211. 16

work page 2001