arxiv: 2605.01652 · v1 · submitted 2026-05-03 · ⚛️ physics.plasm-ph

Recognition: unknown

Computational boundary specification in 3D fixed-boundary magnetohydrodynamic equilibrium modeling

Authors on Pith no claims yet

Pith reviewed 2026-05-09 17:14 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph

keywords MHD equilibriafixed-boundary solver3D plasma modelingstellaratorscomputational boundarymagnetohydrodynamicsedge modeling

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The pith

Fixed-boundary 3D MHD equilibrium solvers can use general computational boundaries in vacuum regions where current and pressure vanish.

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

The paper argues that computational boundaries for 3D fixed-boundary MHD equilibria should be placed outside the plasma edge transition layer in a vacuum-like region where current and pressure drop to zero. This setup allows the boundary to have a nonzero normal magnetic field component, non-constant pressure, and nonzero normal current, conditions that existing tools typically avoid by placing the boundary inside the edge. The authors derive an algorithm that makes such a general-boundary solver possible and note that coil optimization routines for stellarators would then need to match free-boundary equilibrium requirements. A sympathetic reader would care because the change supports more realistic modeling of toroidal plasmas without forcing artificial assumptions at the edge.

Core claim

The authors derive an algorithm for a fixed-boundary 3D magnetohydrodynamic equilibrium solver compatible with a general computational boundary satisfying B · n ≠ 0, p ≠ constant, and J × n ≠ 0, achieved by locating the boundary exterior to the plasma edge transition layer in a region where J and p are approximately zero.

What carries the argument

The derived algorithm that formulates fixed-boundary 3D equilibrium equations on general surfaces without requiring the boundary to be a flux surface or an isobar.

Load-bearing premise

The MHD equilibrium equations remain numerically stable and accurate when solved on a boundary that is not required to be a magnetic flux surface or a surface of constant pressure.

What would settle it

A numerical implementation of the algorithm on a test geometry with a non-flux-surface boundary that fails to converge or violates the force-balance condition J × B = ∇p would show the approach does not hold.

Figures

Figures reproduced from arXiv: 2605.01652 by Alan Kaptanoglu, Tobias Blickhan.

**Figure 2.** Figure 2: Different choices of computational boundary. view at source ↗

**Figure 1.** Figure 1: Representative p and J profiles. J|∂Ω and p|∂Ω approximately vanish on the outer surface view at source ↗

read the original abstract

Outside the core of the plasma, the plasma current and pressure rapidly transition to zero in a scrape-off or edge region or plasma-vacuum interface. However, existing tools for fixed-boundary magnetohydrodynamic equilibria in 2D and 3D domains $\Omega$ typically prescribe the computational boundary $\partial\Omega$ interior to this transition layer. We (1) argue that a more realistic and robust assumption is to define the computational boundary exterior to this transition layer, in a vacuum-like region where $J|_{\partial\Omega} \sim p|_{\partial\Omega} \sim 0$, (2) show that, without this boundary change, existing coil optimization routines for 3D toroidal equilibria (stellarators) should be changed to match free-boundary equilibrium requirements, and (3) derive an algorithm for a fixed-boundary 3D equilibrium solver compatible with a very general computational boundary, with conditions $B \cdot n|_{\partial\Omega} \neq 0$ (not necessarily a flux surface), $p|_{\partial\Omega} \neq \text{const.}$ (not necessarily an isobar), and $J \times n|_{\partial\Omega} \neq 0$.

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

The paper derives an algorithm for fixed-boundary 3D MHD on general vacuum-region boundaries but leaves numerical stability and accuracy unverified.

read the letter

The main takeaway is that they derive an algorithm allowing fixed-boundary 3D MHD solvers to use computational boundaries placed outside the plasma edge in a vacuum-like region. This setup permits B · n ≠ 0, non-constant p, and J × n ≠ 0 on the boundary while still aiming to satisfy the equilibrium equations. They also note the consequences for coil optimization routines in stellarators if the boundary stays interior to the edge layer. The physical motivation for the exterior placement is standard but stated clearly here, and the derivation itself appears to be the concrete new piece. That part is useful because it directly addresses a practical constraint in current 3D equilibrium tools. The connection to coil design is straightforward and relevant for fusion modeling work. The soft spot is verification. The central claim rests on the solver remaining stable and accurate under these relaxed boundary conditions, yet the available description supplies no test cases, error metrics, or comparisons against free-boundary results. Without those checks it is hard to judge whether small nonzero J and p on the boundary introduce spurious modes or degrade the solution. That gap is real but fixable in revision rather than fatal. This paper is for people who build or apply 3D MHD equilibrium codes, especially those optimizing stellarator coils. Readers focused on boundary handling and numerical methods will extract the most value. It deserves a serious referee because the idea targets a genuine limitation in existing workflows and supplies a specific algorithm, even if the implementation evidence needs strengthening. I would send it to peer review and ask for validation examples.

Referee Report

2 major / 2 minor

Summary. The paper argues that computational boundaries for 3D fixed-boundary MHD equilibrium solvers should be placed exterior to the plasma edge transition layer in a vacuum-like region where J and p are approximately zero. It claims this is more realistic than interior placement, shows that existing coil optimization routines for stellarators must be adjusted to match free-boundary requirements without this change, and derives an algorithm for a fixed-boundary solver that accommodates general boundaries satisfying B · n ≠ 0 (not a flux surface), p ≠ constant (not an isobar), and J × n ≠ 0 on ∂Ω while still solving the MHD equilibrium equations.

Significance. If the derived algorithm is numerically stable and accurate, the work could improve realism in stellarator and other 3D toroidal equilibrium modeling by allowing vacuum-region boundaries, with direct implications for coil optimization and free-boundary consistency. The explicit handling of non-standard boundary conditions is a potential strength if validated.

major comments (2)

[§3] The derivation of the algorithm (section 3) must explicitly demonstrate how the boundary-value problem is formulated and discretized to enforce ∇p = J × B, ∇·B = 0, and ∇·J = 0 to high accuracy when p and J are small but nonzero on ∂Ω; without test cases or error metrics, the claim that the solver remains stable for B · n ≠ 0 remains unverified.
[§2] The argument in §2 that coil optimization routines require modification to match free-boundary requirements relies on the boundary shift; however, no quantitative comparison (e.g., difference in optimized coil currents or equilibrium error) is provided to show the practical impact.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a brief statement of the key equations or boundary conditions used in the new algorithm to make the central claim more concrete.
[Introduction] Notation for the computational domain Ω and boundary ∂Ω should be introduced consistently with standard MHD literature to avoid ambiguity in the general boundary conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. Their comments highlight important aspects of clarity and validation that we will address in the revision. We respond to each major comment below.

read point-by-point responses

Referee: [§3] The derivation of the algorithm (section 3) must explicitly demonstrate how the boundary-value problem is formulated and discretized to enforce ∇p = J × B, ∇·B = 0, and ∇·J = 0 to high accuracy when p and J are small but nonzero on ∂Ω; without test cases or error metrics, the claim that the solver remains stable for B · n ≠ 0 remains unverified.

Authors: We agree that greater explicitness on the formulation and discretization is warranted. Section 3 begins from the MHD equilibrium equations and incorporates the general boundary conditions (B · n ≠ 0, p not constant, J × n ≠ 0) by retaining the corresponding surface integrals in the weak form of the variational principle used to derive the algorithm. Divergence-free constraints on B and J are enforced through the choice of finite-element spaces (e.g., H(div)-conforming elements or vector-potential representations), while force balance is satisfied iteratively within the solver. When p and J are small but nonzero on ∂Ω, these surface terms are kept rather than approximated as zero, preserving consistency with the interior equations. We acknowledge that the present manuscript is primarily a derivation and does not contain numerical test cases or quantitative error metrics. In the revised version we will expand Section 3 with a dedicated subsection on the discrete formulation and add a minimal numerical illustration (e.g., a low-resolution test equilibrium) together with basic residual and stability diagnostics for the B · n ≠ 0 case. A full convergence study and extensive benchmarking lie beyond the scope of this theoretical contribution and will be pursued separately. revision: partial
Referee: [§2] The argument in §2 that coil optimization routines require modification to match free-boundary requirements relies on the boundary shift; however, no quantitative comparison (e.g., difference in optimized coil currents or equilibrium error) is provided to show the practical impact.

Authors: Section §2 demonstrates analytically that a fixed-boundary equilibrium computed inside the edge transition layer produces a mismatch with free-boundary coil requirements because the neglected edge currents and pressure gradients alter the normal-field boundary condition seen by the coils. The mismatch is expressed in terms of the integrated edge current and the radial shift of the computational boundary. While we do not present a full coil-optimization run, the derivation already supplies an order-of-magnitude estimate of the resulting discrepancy in the normal field. To strengthen the practical relevance, the revised manuscript will include a short quantitative example based on a representative stellarator equilibrium, showing the approximate change in coil currents or the residual normal-field error that arises if the optimization is not adjusted for the exterior boundary. This addition will be kept concise and will not alter the conceptual argument of the section. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but derivation of general-boundary algorithm remains independent

full rationale

The paper derives an algorithm for fixed-boundary 3D MHD equilibria on domains where B·n ≠ 0, p is non-constant, and J×n ≠ 0 by extending the standard force-balance and divergence-free conditions to a vacuum-like exterior boundary. This extension is presented as a direct mathematical construction from the MHD equations without renaming fitted quantities as predictions or relying on self-citations for the core uniqueness or stability claims. The boundary-placement argument draws from standard plasma-edge descriptions rather than self-referential fitting. One minor self-citation may exist in the broader context but is not load-bearing for the central derivation, yielding a low circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard MHD equilibrium assumptions and the physical description of the plasma scrape-off layer; no free parameters or new entities are mentioned in the abstract.

axioms (2)

standard math MHD force balance and Maxwell equations hold for equilibrium states
Implicit foundation of all fixed-boundary MHD modeling referenced in the abstract.
domain assumption Plasma current and pressure transition rapidly to zero outside the core in a scrape-off or edge region
Explicitly stated as the physical basis for preferring an exterior computational boundary.

pith-pipeline@v0.9.0 · 5510 in / 1304 out tokens · 68302 ms · 2026-05-09T17:14:22.902754+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

81 extracted references · 8 canonical work pages · 1 internal anchor

[1]

physical pressure

Monotonic decrease of the energy. With these boundary conditions, we define the mag- netic energyε B :H curl,div 0 →Rof the fieldBby εB := Z Ω |B|2 2 dV= 1 2 ∥B∥2 L2(Ω).(A1) SinceB 0 is fixed in time, we can take the variation and obtain, δεB = Z Ω B·δB dV= Z Ω B· ∇ ×(v×B)dV,(A2) wherevdenotes the ideal-relaxation velocity. Using Z Ω B·(∇ ×E)dV= Z Ω (∇ ×B...
[2]

In non-contractible domains where the magnetic field has a non-zero normal boundary traceg 0, we introduce a static, curl-free reference fieldB ref such thatg 0 =B ref · n|∂Ω

Modified relative helicity balance. In non-contractible domains where the magnetic field has a non-zero normal boundary traceg 0, we introduce a static, curl-free reference fieldB ref such thatg 0 =B ref · n|∂Ω. The increment field is defined as ˜B=B−B ref, which satisfies ˜B·n| ∂Ω = 0. The fields are decomposed into their rotational and harmonic parts:B=...
[3]

Grad and H

H. Grad and H. Rubin, inProceedings of the 2nd United Nations Conference on the Peaceful Uses of Atomic En- ergy, Vol. 31 (1958) pp. 190–197

1958
[4]

V. D. Shafranov, inReviews of Plasma Physics, Vol. 2 (1966) pp. 103–151

1966
[5]

J. P. Freidberg,Ideal Magnetohydrodynamics(Plenum Press, New York, 1987)

1987
[6]

Guazzotto and J

L. Guazzotto and J. P. Freidberg, Journal of Plasma Physics87, 905870303 (2021)

2021
[7]

Guazzotto and J

L. Guazzotto and J. P. Freidberg, Journal of Plasma Physics87, 905870305 (2021)

2021
[8]

Fitzpatrick, Physics of Plasmas31, 082505 (2024)

R. Fitzpatrick, Physics of Plasmas31, 082505 (2024)

2024
[9]

L. S. Solov’ev, Soviet Physics JETP26, 400 (1968), en- glish translation of Zh. Eksp. Teor. Fiz. 53, 626–643 (1967)

1968
[10]

A. J. Cerfon and J. P. Freidberg, Physics of Plasmas17, 032502 (2010)

2010
[11]

Guazzotto, R

L. Guazzotto, R. Betti, J. Manickam, and S. Kaye, Physics of Plasmas14, 112508 (2007)

2007
[12]

Pataki, A

A. Pataki, A. J. Cerfon, J. P. Freidberg, L. Greengard, and M. O’Neil, Journal of Computational Physics243, 28 (2013)

2013
[13]

T. Xu, R. Fitzpatrick, and F. L. Waelbroeck, Nuclear Fusion59, 064002 (2019)

2019
[14]

L. L. Lao, H. E. St. John, R. D. Stambaugh, A. G. Kell- man, and W. Pfeiffer, Nuclear Fusion25, 1611 (1985). 11

1985
[15]

L. L. Lao, H. E. St. John, Q. Peng, J. R. Ferron, E. J. Strait, T. S. Taylor, W. H. Meyer, C. Zhang, and K. I. You, Fusion Science and Technology48, 968 (2005)

2005
[16]

J. R. Ferron, M. L. Walker, L. L. Lao, H. E. St. John, D. A. Humphreys, and J. A. Leuer, Nuclear Fusion38, 1055 (1998)

1998
[17]

Moret, B

J.-M. Moret, B. P. Duval, H. B. Le, S. Coda, F. Felici, and H. Reimerdes, Fusion Engineering and Design91, 1 (2015)

2015
[18]

Landreman, H

M. Landreman, H. M. Smith, A. Moll´ en, and P. Helander, Physics of Plasmas21, 042503 (2014)

2014
[19]

Landreman, S

M. Landreman, S. Buller, and M. Drevlak, Physics of Plasmas29, 082501 (2022)

2022
[20]

Bader, B

A. Bader, B. J. Faber, J. C. Schmitt, D. T. Ander- son, M. Drevlak, J. M. Duff, H. Frerichs, C. C. Hegna, T. G. Kruger, M. Landreman, I. J. McKinney, L. Singh, J. M. Schroeder, P. W. Terry, and A. S. Ware, Journal of Plasma Physics86, 905860506 (2020)

2020
[21]

Reiman and H

A. Reiman and H. Greenside, Computer Physics Com- munications43, 157 (1986)

1986
[22]

S. R. Hudson, D. A. Monticello, A. H. Reiman, A. H. Boozer, D. J. Strickler, S. P. Hirshman, and M. C. Zarn- storff, Physical Review Letters89, 275003 (2002)

2002
[23]

Kanno, T

R. Kanno, T. Hayashi, N. Takeuchi, K. Ichiguchi, and N. Nakajima,Computational Study of Three Dimensional MHD Equilibrium and Stability with Magnetic Islands, Tech. Rep. NIFS-767 (National Institute for Fusion Sci- ence, 2002)

2002
[24]

Suzuki, N

Y. Suzuki, N. Nakajima, K. Watanabe, Y. Nakamura, and T. Hayashi, Nuclear Fusion46, L19 (2006)

2006
[25]

S. R. Hudson, R. L. Dewar, G. Dennis, M. J. Hole, M. McGann, G. von Nessi, and S. Lazerson, Physics of Plasmas19, 112502 (2012)

2012
[26]

Enciso, A

A. Enciso, A. Luque, and D. Peralta-Salas, Journal of the European Mathematical Society27, 2251 (2025)

2025
[27]

S. P. Hirshman and J. C. Whitson, Physics of Fluids26, 3553 (1983)

1983
[28]

Schilling, The numerics of vmec++ (2025), arXiv:2502.04374

J. Schilling, The numerics of vmec++ (2025), arXiv:2502.04374

work page arXiv 2025
[29]

Hindenlang, G

F. Hindenlang, G. G. Plunk, and O. Maj, Plasma Physics and Controlled Fusion67, 045002 (2025)

2025
[30]

Dudt and E

D. Dudt and E. Kolemen, Physics of Plasmas27(2020)

2020
[31]

Panici, R

D. Panici, R. Conlin, D. W. Dudt, K. Unalmis, and E. Kolemen, Journal of Plasma Physics89, A373 (2023)

2023
[32]

Blickhan, J

T. Blickhan, J. Stratton, and A. A. Kaptanoglu, Mrx: A differentiable 3d mhd equilibrium solver without nested flux surfaces (2025), arXiv:2510.26986

work page arXiv 2025
[33]

Loizu, S

J. Loizu, S. Hudson, A. Bhattacharjee, and P. Helander, Physics of Plasmas22(2015)

2015
[34]

S. A. Lazerson, J. Loizu, S. Hirshman, and S. R. Hudson, Physics of Plasmas23(2016)

2016
[35]

Imbert-G´ erard, E

L.-M. Imbert-G´ erard, E. J. Paul, and A. M. Wright,An introduction to stellarators: From magnetic fields to sym- metries and optimization(SIAM, 2024)

2024
[36]

S. R. Hudson, J. Loizu, C. Zhu, Z. S. Qu, C. N”uhrenberg, S. A. Lazerson, C. B. Smiet, and M. J. Hole, Plasma Physics and Controlled Fusion62, 084002 (2020)

2020
[37]

Baillod, A

A. Baillod, A. Kumar, J. Loizu, S. R. Hudson, and M. J. Hole, Journal of Plasma Physics87, 905870614 (2021)

2021
[38]

S. R. Hudson, D. Panici, C. Zhu, G. S. Wood- bury Saudeau, A. Baillod, M. Cianciosa, and A. S. Ware, Physics of Plasmas32, 012505 (2025)

2025
[39]

S. P. Hirshman, R. Sanchez, and C. R. Cook, Physics of Plasmas18, 062504 (2011)

2011
[40]

Peraza-Rodriguez, J

H. Peraza-Rodriguez, J. M. Reynolds-Barredo, R. Sanchez, J. Geiger, V. Tribaldos, S. P. Hirsh- man, and M. Cianciosa, Physics of Plasmas24, 082516 (2017)

2017
[41]

H. A. Peraza Rodriguez,Free-boundary extension of the SIESTA code and its application to the Wendelstein 7- X stellarator, Master’s thesis, Universidad Carlos III de Madrid (2017)

2017
[42]

Suzuki, Plasma Physics and Controlled Fusion59, 054008 (2017)

Y. Suzuki, Plasma Physics and Controlled Fusion59, 054008 (2017)

2017
[43]

Chodura and A

R. Chodura and A. Schl¨ uter, Journal of Computational Physics41, 68 (1981)

1981
[44]

W. Park, D. A. Monticello, H. Strauss, and J. Manickam, The Physics of Fluids29, 1171 (1986)

1986
[45]

Arnold, R

D. Arnold, R. Falk, and R. Winther, Bulletin of the American Mathematical Society47, 281 (2010)

2010
[46]

Conlin, J

R. Conlin, J. Schilling, D. W. Dudt, D. Panici, R. Jorge, and E. Kolemen, High order free boundary mhd equilib- ria in desc (2024), arXiv:2412.05680

work page arXiv 2024
[47]

Nikulsin, R

N. Nikulsin, R. Ramasamy, M. Hoelzl, F. Hindenlang, E. Strumberger, K. Lackner, S. G¨ unter, J. Team,et al., Physics of Plasmas29(2022)

2022
[48]

Y. Zhou, N. M. Ferraro, A. M. Wright, A. Reiman, R. Jorge, S. P. Smith, M. Landreman, M. Dudt, and D. A. Gates, Physics of Plasmas30, 032503 (2023)

2023
[49]

Patil and C

S. Patil and C. Sovinec, inAPS Division of Plasma Physics Meeting Abstracts, Vol. 2023 (2023) pp. GP11– 116

2023
[50]

K. Hu, Y. Ma, and J. Xu, Numerische Mathematik135, 371 (2017)

2017
[51]

E. S. Gawlik and F. Gay-Balmaz, Journal of Computa- tional Physics450, 110847 (2022)

2022
[52]

Carlier and M

V. Carlier and M. Campos-Pinto, Variational dis- cretizations of ideal magnetohydrodynamics in smooth regime using finite element exterior calculus (2024), arXiv:2402.02905

work page arXiv 2024
[53]

M. He, P. E. Farrell, K. Hu, and B. D. Andrews, Topology-preserving discretization for the magneto- frictional equations arising in the parker conjecture (2025), arXiv:2501.11654

work page arXiv 2025
[54]

J. C. Schmitt, D. T. Anderson, E. C. Andrew, A. Bader, K. Camacho Mata, J. M. Canik, L. Carbajal, A. Cer- fon, W. A. Cooper, N. M. Davila, W. D. Dorland, J. M. Duff, W. Guttenfelder, C. C. Hegna, D. P. Huet, M. Lan- dreman, G. Le Bars, A. Malkus, N. R. Mandell,et al., Journal of Plasma Physics91, e88 (2025)

2025
[55]

J. D. Hanson, Plasma Physics and Controlled Fusion57, 115006 (2015)

2015
[56]

Toler, A

E. Toler, A. Cerfon, and D. Malhotra, Journal of Plasma Physics90, 175900202 (2024)

2024
[57]

Y. Zhou, N. M. Ferraro, D. M. Walker, S. P. Smith, S. D. Hudson, and A. H. Boozer, Nuclear Fusion61, 086015 (2021)

2021
[58]

Merkel, Nuclear Fusion27, 867 (1987)

P. Merkel, Nuclear Fusion27, 867 (1987)

1987
[59]

Landreman, Nuclear Fusion57, 046003 (2017)

M. Landreman, Nuclear Fusion57, 046003 (2017)

2017
[60]

S. A. Henneberg, S. R. Hudson, D. Pfefferl´ e, and P. He- lander, Journal of Plasma Physics87, 905870226 (2021)

2021
[61]

Ku and A

L.-P. Ku and A. H. Boozer, Physics of Plasmas17, 122503 (2010)

2010
[62]

A. H. Boozer, Journal of Plasma Physics81, 515810606 (2015)

2015
[63]

L. Fu, E. J. Paul, A. A. Kaptanoglu, and A. Bhattachar- jee, Nuclear Fusion65, 026045 (2025). 12

2025
[64]

Panici, R

D. Panici, R. Conlin, R. Gaur, D. W. Dudt, Y. G. El- macioglu, M. Landreman, T. Elder, N. Snir, I. Gissis, Y. Nikulshin, and E. Kolemen, Surface current optimiza- tion and coil-cutting algorithms for stage-two stellarator optimization (2025), arXiv:2508.09321

work page arXiv 2025
[65]

L. Fu, D. Panici, E. Paul, A. Kaptanoglu, and A. Bhattacharjee, A flexible and differentiable coil proxy for stellarator equilibrium optimization (2025), arXiv:2510.16243

work page internal anchor Pith review Pith/arXiv arXiv 2025
[66]

C. Zhu, S. R. Hudson, Y. Song, and Y. Wan, Nuclear Fusion58, 016008 (2018)

2018
[67]

Landreman, B

M. Landreman, B. Medasani, F. Wechsung, A. Giuliani, R. Jorge, and C. Zhu, Journal of Open Source Software 6, 3525 (2021)

2021
[68]

S. R. Hudson, C. Zhu, D. Pfefferl´ e, and L. Gunderson, Physics Letters A382, 2732 (2018)

2018
[69]

A. A. Kaptanoglu, A. Wiedman, J. Halpern, S. Hur- witz, E. J. Paul, and M. Landreman, Nuclear Fusion65, 046029 (2025)

2025
[70]

P. F. Gil, W. Li, J. Stratton, A. A. Kaptanoglu, and E. V. Stenson, Augmented lagrangian methods produce cutting-edge magnetic coils for stellarator fusion reactors (2025), arXiv:2507.12681

work page arXiv 2025
[71]

X. Nie, J. Peng, Y. Xie, G. Yu, K. Liu, and C. Zhu, Nuclear Fusion65, 086008 (2025)

2025
[72]

Jorge, M

R. Jorge, M. Landreman, and P. Helander, Plasma Physics and Controlled Fusion66, 074002 (2024)

2024
[73]

Constantin, T

P. Constantin, T. D. Drivas, and D. Ginsberg, Nonlin- earity35, 2363 (2022)

2022
[74]

Wiegmann, Y

C. Wiegmann, Y. Suzuki, D. Reiter, and R. C. Wolf, in 36th EPS Conference on Plasma Physics(2009) eCA Vol. 33E, P-1.170

2009
[75]

Suzuki, K

Y. Suzuki, K. Y. Watanabe,et al., inPlasma Conference 2014(2014) poster 18PB-055

2014
[76]

Suzuki, K

Y. Suzuki, K. Y. Watanabe, and S. Sakakibara, Physics of Plasmas27, 102502 (2020)

2020
[77]

H. C. Brinkman, Applied Scientific Research1, 27 (1949)

1949
[78]

Temam,Navier–Stokes Equations: Theory and Nu- merical Analysis(North-Holland, Amsterdam, 1977)

R. Temam,Navier–Stokes Equations: Theory and Nu- merical Analysis(North-Holland, Amsterdam, 1977)

1977
[79]

Girault and P.-A

V. Girault and P.-A. Raviart,Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms (Springer-Verlag, Berlin, 1986)

1986
[80]

Boffi, F

D. Boffi, F. Brezzi, and M. Fortin,Mixed Finite Element Methods and Applications(Springer, Berlin, 2013)

2013

Showing first 80 references.