Self-adjoint curl operators

Hiptmair, Ralf; Kotiuga, P. Robert; Tordeux, Sébastien

doi:10.1007/s10231-011-0189-y

Cited by 22 publications

(41 citation statements)

References 32 publications

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“…Then we define curl of a vector field X, denoted by ∇ × X, by ∇ × X = ( * dι X g) ♯ [18,19]. For the functional-analytic and spectral treatment of curl operator, see [25]. In this paper we deal with the most basic example M = R 3 , and the standard Euclidean metric.…”

Section: The Main Resultsmentioning

confidence: 99%

Beltrami vector fields with an icosahedral symmetry

Alkauskas

2020

Journal of Geometry and Physics

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Section: The Main Resultsmentioning

confidence: 99%

Beltrami vector fields with an icosahedral symmetry

Alkauskas

2020

Journal of Geometry and Physics

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“…We consider a one-dimensional Klein-Gordon type equation for one component of the electric field with periodic boundary conditions on the interval [− 10,14], where the plasma occupies the region (10,11). This equation is obtained by eliminating b and p from (1).…”

Section: Klein-gordon Type Equation -Two Step Methodsmentioning

confidence: 99%

“…Equation (1) has to be supplemented with boundary conditions and initial values. The theory developed below applies to the case of perfect magnetic conductor (PMC), perfect electric conductor (PEC) or periodic boundary conditions, which guarantee that the "curl curl" operator is self-adjoint [10].…”

Section: Physical Problem and Spatial Discretizationmentioning

confidence: 99%

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Convergence analysis of an explicit splitting method for laser plasma interaction simulations

Jansing¹,

Schädle²

2018

etna

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Abstract. The convergence of a triple splitting method originally proposed by Tückmantel, Pukhov, Liljo, and Hochbruck for the solution of a simple model that describes laser plasma interactions with overdense plasmas is analyzed. For classical explicit integrators it is the large density parameter that imposes a restriction on the time step size to make the integration stable. The triple splitting method contains an exponential integrator in its central component and was specifically designed for systems that describe laser plasma interactions and overcomes this restriction. We rigorously analyze a slightly generalized version of the original method. This analysis enables us to identify modifications of the original scheme such that a second-order convergent scheme is obtained.Key words. exponential integrators, highly oscillatory problems, trigonometric integrators, splitting methods AMS subject classifications. 65P101. Introduction. We consider the numerical solution of a system of equations describing laser plasma interactions with an overdense plasma that is a simplified version of the models considered in [14,16]. Essentially as in [17], the laser is described by Maxwell's equations, and the plasma is modeled as a fluid only, in contrast to [14,16]. After discretizing in space with a fixed spatial grid size h, a system of ordinary differential equations is obtained. This highly oscillatory system of ordinary differential equations is discretized in time by a triple splitting method with filter functions. The introduction of filter functions is a widely-used method to avoid resonance effects in splitting methods applied to oscillatory differential equations; see, e.g., [4, 6, 7, 11] and [8].The situation to consider now is slightly unusual as it is the localized overdense plasma and not the spatial discretization that gives rise to fast oscillations in the solution. The plasma frequency is several orders of magnitude larger than the laser frequency, which has to be resolved by the spatial grid. Hence, it is the plasma density ρ that imposes a step size restriction in explicit Runge-Kutta or multistep methods. To overcome the restriction on the time step size due to the plasma density, a triple splitting method with filter functions was introduced in [13,17] for this model problem. An astute choice of filter functions results in a method that shows excellent behavior in numerical experiments. The numerical experiments in [17] indicate convergence of second order in the time step size τ independent of ρ. A more detailed experiment, which is reported in Section 8.1, reveals that the method from [17] is not of second order in τ independent of the plasma density but is merely stable.By our convergence analysis of the triple splitting we are able to formulate conditions on the filter functions to obtain second-order convergence in τ independent of the plasma density ρ. These conditions can be fulfilled by slightly modifying the choice of the filter functions originally proposed in [13,17]. The modification comes ...

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“…Inspired by the theory of self-adjoint curl operators in dimension 3 (a long story, stretching from [22] to [23]), we observe that Lemma 4.3. BS is a self-adjoint operator on closed Dirichlet (k + 1)-forms on Ω, for odd k. For any two such forms α and β,…”

Section: Constructing a Primitive Formmentioning

confidence: 95%

A new cohomological formula for helicity inR2k+1reveals the effect of a diffeomorphism on helicity

Cantarella

Parsley

2010

Journal of Geometry and Physics

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a b s t r a c tThe helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed (k + 1)-forms on a domain in (2k+1)-space that vanish when pulled back to the boundary of the domain.Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot-Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard 'folk' theorems about the cohomology of the boundary of domains in R 2k+1 .

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Self-adjoint curl operators

Cited by 22 publications

References 32 publications

Beltrami vector fields with an icosahedral symmetry

Beltrami vector fields with an icosahedral symmetry

Convergence analysis of an explicit splitting method for laser plasma interaction simulations

A new cohomological formula for helicity inR2k+1reveals the effect of a diffeomorphism on helicity

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