A nonlinear relaxation formulation of the p-curl problem modelling high-temperature superconductors: A modified Yee's scheme

Law, Yann-Meing; Laforest, Marc

doi:10.1016/j.jcp.2018.11.027

Cited by 3 publications

(3 citation statements)

References 40 publications

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“…Let us mention as third application the p ‐

CurlCurl

problem 55–57 appearing in modeling the magnetic field in a high‐temperature superconductor. By the definition of the Banach space

W^{1, p} false(Curl; normalΩ, ℝ^{3 \times 3} false)

, it is clear that

Curl false(false‖ Curl P {false‖}^{p - 2} Curl P false) + P = G, P \times ν_{false| \partial normalΩ} = 0

with

G \in L^{p'} false(normalΩ, ℝ^{3 \times 3} false)

admits a unique solution for 1 < p ≤ 2, since () is the Euler–Lagrange equation to the strictly convex minimization problem

\int_{normalΩ} \frac{1}{2} false‖ P {false‖}^{2} + \frac{1}{p} false‖ Curl P {false‖}^{p} - (⟨⟩, G, P) normald x \to \min .

…”

Section: Introductionmentioning

confidence: 99%

“…Let us mention as third application the p-CurlCurl problem [55][56][57] appearing in modeling the magnetic field in a high-temperature superconductor. By the definition of the Banach space W 1, p (Curl; Ω, R 3×3 ), it is clear that Curl (||Curl P|| p−2 Curl P) + P = G, P × 𝜈 |𝜕Ω = 0 (14) with G ∈ L p′ (Ω, R 3×3 ) admits a unique solution for 1 < p ≤ 2, since ( 14) is the Euler-Lagrange equation to the strictly convex minimization problem…”

Section: Introductionmentioning

confidence: 99%

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Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields

Lewintan

Neff

2021

Math Methods in App Sciences

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For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in W01,0.1empfalse(Curl;normalΩ,ℝ3×3false). More precisely, let normalΩ⊂ℝ3 be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) > 0 such that ‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3) holds for all tensor fields P∈W01,0.1empfalse(Curl;normalΩ,ℝ3×3false), that is, for all P∈W1,0.1empfalse(Curl;normalΩ,ℝ3×3false) with vanishing tangential trace P×ν=0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible P=normalDu, this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric P=A an Lp‐version of the Poincaré inequality.

show abstract

“…Let us mention as third application the p ‐

CurlCurl

problem 55–57 appearing in modeling the magnetic field in a high‐temperature superconductor. By the definition of the Banach space

W^{1, p} false(Curl; normalΩ, ℝ^{3 \times 3} false)

, it is clear that

Curl false(false‖ Curl P {false‖}^{p - 2} Curl P false) + P = G, P \times ν_{false| \partial normalΩ} = 0

with

G \in L^{p'} false(normalΩ, ℝ^{3 \times 3} false)

admits a unique solution for 1 < p ≤ 2, since () is the Euler–Lagrange equation to the strictly convex minimization problem

\int_{normalΩ} \frac{1}{2} false‖ P {false‖}^{2} + \frac{1}{p} false‖ Curl P {false‖}^{p} - (⟨⟩, G, P) normald x \to \min .

…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields

Lewintan

Neff

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Let us mention as third application the p-Curl Curl problem [53,55,67] appearing in modeling the magnetic field in a high-temperature superconductor. By the definition of the Banach space…”

Section: Introductionmentioning

confidence: 99%

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

Lewintan,

Neff

2019

Preprint

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3-D Numerical Modeling for the Magnetization of Superconductors Using a Local Discontinuous Galerkin Finite Element Method

2019

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A nonlinear relaxation formulation of the p-curl problem modelling high-temperature superconductors: A modified Yee's scheme

Cited by 3 publications

References 40 publications

Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields

Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

3-D Numerical Modeling for the Magnetization of Superconductors Using a Local Discontinuous Galerkin Finite Element Method

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A nonlinear relaxation formulation of the p-curl problem modelling high-temperature superconductors: A modified Yee's scheme

Cited by 3 publications

References 40 publications

Nečas–Lions lemma revisited: An Lp‐version of the generalized Korn inequality for incompatible tensor fields

Nečas–Lions lemma revisited: An Lp‐version of the generalized Korn inequality for incompatible tensor fields

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

3-D Numerical Modeling for the Magnetization of Superconductors Using a Local Discontinuous Galerkin Finite Element Method

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About

Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields

Nečas–Lions lemma revisited: An L^p‐version of the generalized Korn inequality for incompatible tensor fields