On Friedrichs Inequality, Helmholtz Decomposition, Vector Potentials, and the div-curl Lemma

Schweizer, Ben

doi:10.1007/978-3-319-75940-1_4

Cited by 29 publications

(32 citation statements)

References 12 publications

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“…But in our case (namely when the bilinear form has a trivial kernel), the coercivity of a can be concluded also in the classical sense, i.e. ψ 2 H 1 ≤ C a(ψ, ψ) for all ψ ∈ X, compare [26].…”

Section: A Construction Of Vector Potentialsmentioning

confidence: 73%

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Lipton

Schweizer

2018

Arch Rational Mech Anal

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Section: A Construction Of Vector Potentialsmentioning

confidence: 73%

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Lipton

Schweizer

2018

Arch Rational Mech Anal

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“…For a proof of the global div-curl lemma without t-dependence see e.g. Lemma 6.1 in [27]. For the case with t-dependence as in (3.24), we argue as follows.…”

Section: Proofmentioning

confidence: 99%

Strain gradient visco-plasticity with dislocation densities contributing to the energy

Röger

Schweizer

2017

Math. Models Methods Appl. Sci.

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We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.

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“…Suppose that nullspace consistent tensor product SBP operators are applied in three space dimensions. Then, (49) and the kernel of the discrete curl operator can be decomposed into the direct orthogonal sum…”

Section: Three Space Dimensions Revisitedmentioning

confidence: 99%

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

Ranocha

Ostaszewski

Heinisch

2020

Commun. Appl. Math. Comput.

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In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated to grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully.In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first order partial differential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

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On Friedrichs Inequality, Helmholtz Decomposition, Vector Potentials, and the div-curl Lemma

Abstract: We study connections between four different types of results that are concerned with vector-valued functions u :Coercivity results in H 1 (Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma.

Cited by 29 publications

References 12 publications

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Effective Maxwell’s Equations for Perfectly Conducting Split Ring Resonators

Strain gradient visco-plasticity with dislocation densities contributing to the energy

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

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