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Topic: Helmholtz decomposition

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	Helmholtz decomposition - Wikipedia, the free encyclopedia
		In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any vector field, meeting certain conditions (of decay towards infinity) can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields.
		The resulting Helmholtz decomposition of a vector field, which is twice continuously differentiable and with rapid enough decay at infinity, splits the vector field into a sum of gradient and curl as follows:
		The decomposition of a differential form is known as the Hodge decomposition.
	en.wikipedia.org /wiki/Helmholtz_decomposition (278 words)

	Encyclopedia: Helmholtz decomposition
		Compatibility decomposition In Unicode, the compatibility decomposition of a character is a string that holds the relati...
		Decomposition Decomposition is the reduction of bodies and other formerly living taphonomy.
		Helmholtz decomposition The Helmholtz decomposition of a vector field which is twice continuously differentiable, with r...
	www.nationmaster.com /encyclopedia/Helmholtz-decomposition (668 words)

	Helmholtz theorem - Wikipedia, the free encyclopedia
		There exist several theorems named after Hermann von Helmholtz.
		Helmholtz theorem in vector calculus, also known as fundamental theorem of vector calculus; see Helmholtz decomposition.
		Helmholtz theorem in classical mechanics; see Helmholtz theorem (classical mechanics).
	en.wikipedia.org /wiki/Helmholtz_theorem (108 words)

	Helmholtz
		When Faraday began to study the phenomena of decomposition by the galvanic current, which of course were considered by Berzelius as among the firmest supports of his theory, he put a very simple question- the first question, indeed, which every chemist speculating about electrolysis ought to have thought of.
		Then he compared the amount of decomposition in cells containing different electrolytes, and he found it exactly proportional to the chemical equivalents of the elements, which were either separated or converted into new compounds.
		Conversely, a decomposition of the compound substance requires work, because it restores the energy, for when the hydrogen is burned in the oxygen, they unite, form water, and develop a great amount of heat.
	dbhs.wvusd.k12.ca.us /webdocs/Chem-History/Helmholtz-1881.html (9718 words)

	SISC Volume 20 Issue 5
		The two-dimensional Helmholtz equation for problems where the physical domain consists of layers with different material properties is studied.
		A domain decomposition method is used, in which the preconditioning is based on the Schur complement algorithm with "fast Poisson-type" preconditioners for the subdomains.
		The memory requirements for the preconditioner are nearly linear in the number of unknowns.
	epubs.siam.org /sam-bin/dbq/article/32532 (286 words)

	LORENE: Vector class Reference
		Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.
		Returns the div-free vector in the Helmholtz decomposition.
		Assigns a new vectorial basis (triad) of decomposition.
	www.lorene.obspm.fr /Refguide/classVector.html (1798 words)

	Domain Decomposition for Helmholtz Problems and Algorithms for Parallel Computing (Site not responding. Last check: 2007-10-15)
		The Helmholtz equation is the basic equation governing the propagation and scattering of time harmonic sound.
		The subdomain interface conditions are updated using a Domain Decomposition iterative scheme originating in the Schwarz alternating method.
		This Domain Decomposition methodology is implemented using Finite Element discretization on unstructured meshes.
	www.nd.edu /~cam/seminars/lectures/Resiga.htm (239 words)

	Ingber Research Page (Site not responding. Last check: 2007-10-15)
		An animation from a vorticity code based on the generalized Helmholtz decomposition
		"A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations," (with C. Chen and J. Tanski), Int.
		"Parallel multipole implementation of the generalized Helmholtz decomposition for solving viscous flow problems," (with M. Brown and A. Mammoli), Int.
	www.me.unm.edu /~ingber/research.html (402 words)

	Nat' Academies Press, Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994)
		When the decomposition of the pressure field was described in the talk, it was mentioned that this decomposition might be useful in Reynolds-averaged Navier-Stokes computations about ships.
		A Helmholtz decomposition provides a rigorous treatment of the intersection of the ship hull with the free surface.
		A Helmholtz formulation also allows the use of boundary -layer approximations in the free-surface boundary conditions.
	www.nap.edu /openbook/NI000061/html/270.html (631 words)

	bibliography.bib
		Contains a new Helmholtz-like decomposition result.} } @ARTICLE{em-dhl-eig, AUTHOR = {Dhia, Anne-Sophie B. and Hazard, Christophe and Lohrengel, Stephanie}, TITLE = { A Singular Field Method for the Solution of {M}axwell's Equations in Polyhedral Domains }, JOURNAL = {SIAM J. Appl.
		Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements.
		A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly.
	www.isc.tamu.edu /~tkolev/papers/_thesis_/bibliography/bibliography-bib.html (2900 words)

	Project-Team - VISTA
		It involves a data-driven term, deduced from the integration of the continuity equation between two successive images, and a regularization term based on the Helmholtz decomposition of vector fields.
		These parameters are the blooming parameter which is the derivative of the vorticity magnitude and reveals the evolution of turbulent structures, and the spatial derivative of the absolute motion magnitude which accounts for the speed of the flow.
		Besides, we have extended the described fluid motion estimation method relying on the Helmholtz decomposition to compute the associated potential functions (namely, the stream function and the velocity potential) directly from the image sequence.
	www.inria.fr /rapportsactivite/RA2003/vista/module18.html (2593 words)

	Publications of Dr. Ralf Hiptmair
		Both can be tackled by nodal multilevel decompositions where for the former the splitting is set in the space of discrete scalar potentials.
		Generalized Helmholtz decompositions - orthogonal decompositions into the null space of the relevant differential operator and its complement - are crucial in our considerations.
		This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid method.
	www.sam.math.ethz.ch /~hiptmair/Homepage/abstract.html (9334 words)

	Color - Wikipedia, the free encyclopedia
		Although Aristotle and other ancient scientists speculated on the nature of light and color vision, it was not until Newton that light was correctly identified as the source of the color sensation.
		Goethe studied the theory of colors, and in 1801 Thomas Young proposed his trichromatic theory which was later refined by Hermann von Helmholtz.
		That theory was confirmed in the 1960s and will be described below.
	en.wikipedia.org /wiki/Color (3587 words)

	KLUEDO - Wavelet Analysis of the Geomagnetic Field Within Source Regions of Ionospheric and Magnetospheric Currents
		A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents.
		The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields.
		Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics.
	kluedo.ub.uni-kl.de /volltexte/1999/889/index.html (166 words)

	Overlapping Schwarz Methods for Vector Valued Elliptic Problems in Three Dimensions (Site not responding. Last check: 2007-10-15)
		The emphasis is on a unified framework for the motivation and theoretical study of the various approaches developed in recent years.
		Generalized Helmholtz decompositions -- orthogonal decompositions into the null space of the relevant differential operator and its complement -- are crucial in our considerations.
		It turns out that the decompositions the Schwarz methods are based upon have to be designed separately for both components.
	csdocs.cs.nyu.edu /Dienst/UI/2.0/Describe/ncstrl.nyu_cs%2fTR1997-746 (200 words)

	IngentaConnect A Galerkin Implementation of the Generalized Helmholtz Decomposit... (Site not responding. Last check: 2007-10-15)
		However, the accurate implementation of the boundary conditions seems to continue to be an impediment to the acceptance and use of numerical methods based on vorticity formulations.
		This can be accomplished in one of two ways by either solving for boundary vorticity (leading to a Dirichlet boundary condition for the vorticity equation) or solving for boundary vortex sheet strengths (leading to a Neumann condition).
		To more accurately satisfy both components of the velocity boundary conditions, a Galerkin formulation is applied to the generalized Helmholtz decomposition.
	www.ingentaconnect.com /content/ap/cp/2001/00000169/00000001/art06724 (379 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition.
		The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition.
		The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step.
	zmath.library.ualberta.ca /cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=tex&an=01868351&format=complete (209 words)

	A Domain Decomposition Method for Helmholtz Scattering Problems (ResearchIndex)
		Abstract: Introduction We present a study of iterative nonoverlapping domain decomposition methods (DDMs) for the harmonic scattering wave equation in the 3D case.
		7 emes de propagation d (context) - de d'ecomposition, conditions et al.
		Second-Order Transmission Conditions for the Helmholtz Equation - Douglas, Jr., Meade (1998)
	citeseer.ist.psu.edu /498475.html (309 words)

	Bernhard Hientzsch's research (Site not responding. Last check: 2007-10-15)
		I have developed some domain decomposition preconditioners for spectral element methods for the curl-curl problem appearing in several formulations of Maxwell's equation.
		Since a vector field problem with a non-trivial structure is solved, use and implementation of these methods is also non-trivial, but results in a fast direct solver.
		I also investigated the use of a computational Helmholtz decomposition to solve the curl-curl problem.
	www.cims.nyu.edu /~hientzsc/research.html (279 words)

	SISC Volume 26 Issue 5
		In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci.
		This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique.
		The paper describes in detail the matching algorithm for two boxes which is a basis for the domain decomposition scheme.
	epubs.siam.org /sam-bin/dbq/article/41703 (263 words)

	RMC - Vol. 16 Num. 2, Volker Michel
		Using a standard approach in seismology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy-Navier equation into two non-coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations using the Mie representation.
		Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy-Navier equation.
		Michel, Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth, Rev. Mat.
	www.mat.ucm.es /serv/revista/vol16-2/vol16-2j.html (146 words)

	Algebraic and discrete Velte decompositions (ResearchIndex) (Site not responding. Last check: 2007-10-15)
		Abstract: : Starting from an algebraic equivalent of the Velte decomposition of the function space (H 1 0) n (where n = 2; 3) into three orthogonal subspaces, we consider finite difference and finite element classes for the approximate solution of the first-kind Stokes problem in two and three dimensions which admit discrete Velte decompositions.
		The discrete Velte decomposition should be of particular importance in solving Maxwell equations in the primitive variables in cases when the solutions are...
		8 Towards discrete Velte decompositions and narrow bounds for..
	citeseer.ist.psu.edu /359032.html (348 words)

	Multi-Model Multi-Domain Computational Methods: Papers
		D. Kaushik, D. Keyes and B. Smith, 1998, "On the Interaction of Architecture and Algorithm in the Domain-Based Parallelization of an Unstructured Grid Incompressible Flow Code", in Proceedings of the 10th Intl.
		Our group is active in the international conferences on domain decomposition methods, and three co-PIs have, during this project, assumed editorial responsibilities for these proceedings, as listed chronologically below.
		The annual domain decomposition proceedings have become a prime archive for work related to this project.
	www.cs.odu.edu /~keyes/nsf/papers.html (1943 words)

	Color (Site not responding. Last check: 2007-10-15)
		If we fix (x,y,z) a point in space and look at the solution u(x,y,z,t) as a function of t, we obtain a signal.
		If we take the Fourier transform of this signal, we obtain a frequency decomposition as described above.
		The frequency multiplied by Planck's constant h determines the energy of a photon of the relevant component.
	color.ask.dyndns.dk (4976 words)

	Essam M. Wahba
		My research is concerned with the development of a novel hierarchical formulation for the equations of fluid motion based on a potential function augmented with entropy and vorticity corrections.
		The density and pressure are reformulated in terms of the speed and entropy.
		To quantitatively assess the merits of this new approach, a wide range of inviscid and viscous flow problems are simulated spanning the Mach number range from incompressible to supersonic flows with special emphasis on transonic flows.
	mae.ucdavis.edu /~ewahba (1949 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		Emphasis will be on an efficient iterative solution of the discretized problems by multilevel techniques.
		We note that Helmholtz decompositions are a powerful tool not only in the context of mixed Raviart-Thomas elements but also for Nédélec's edge elements.
		Helmholtz decompositions are a powerful tool not only in the context of mixed Raviart-Thomas elements but also for Nédélec's edge elements.
	www.zarm.uni-bremen.de /gamm98/num_abs/a859.html (362 words)

	SIAP Volume 65 Issue 6
		The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz decomposition of spherical vector fields and (ii) the Mie representation of solenoidal vector fields in terms of poloidal and toroidal parts.
		The wavelet Mie representation is shown to provide an adequate tool for geomagnetic modeling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features.
		To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.
	epubs.siam.org /sam-bin/dbq/article/60379 (224 words)

	Project-Team-VISTA
		The proposed estimator is based on the Helmholtz decomposition of vector fields.
		This decomposition consists in representing the velocity field as the sum of divergence-free component and a curl-free component.
		In order to provide a low-dimensional solution, both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measure.
	www.inria.fr /rapportsactivite/RA2004/vista/uid36.html (927 words)

	SISC Volume 25 Issue 5
		This paper is dedicated to recent developments of a two-Lagrange multipliers domain decomposition method for the Helmholtz equation [C. Farhat et al., Comput.
		Most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator [B. Després, Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman et al., eds., SIAM, Philadelphia, 1993, pp.
		Convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented.
	epubs.siam.org /sam-bin/dbq/article/41535 (326 words)

	Modeling
		He asks whether predictions comparable to those obtained by second-moment closure can be obtained from a simpler closure, if only the mean flow is required.
		The mathematical origin of this question comes from applying the Helmholtz decomposition of vector fields to the RANS equations.
		This is a vector field, to which the Helmholtz decomposition into curl and divergence-free components applies: ${\bf F} = \nabla\phi + \nabla\times\bpsi$.
	ctr.stanford.edu /Summer96/modeling.html (870 words)

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