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| In [[mathematics]], a '''Poincaré–Steklov operator''' (after [[Henri Poincaré]] and [[Vladimir Steklov]]) maps the values of one [[boundary condition]] of the solution of an [[elliptic partial differential equation]] in a [[domain (mathematical analysis)|domain]] to the values of another boundary condition. Usually, either of the boundary conditions determines the solution. Thus, a Poincaré–Steklov operator encapsulates the boundary response of the system modelled by the partial differential equation. When the partial differential equation is discretized, for example by [[finite elements]] or [[finite differences]], the discretization of the Poincaré–Steklov operator is the [[Schur complement]] obtained by eliminating all degrees of freedom inside the domain.
| | I'm Helaine (25) from All Saints South Elmham, United Kingdom. <br>I'm learning Bengali literature at a local university and I'm just about to graduate.<br>I have a part time job in a university.<br><br><br>http://brokerltd.com - CPM<br><br>Look at my page - [http://brokerltd.com publisher] |
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| Note that there may be many suitable different boundary conditions for a given partial differential equation and the direction in which a Poincaré–Steklov operator maps the values of one into another is given only by a convention.<ref name="Bossavit">A. Bossavit, The "scalar" Poincaré–Steklov operator and the "vector" one: algebraic structures which underlie their duality. In ''Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Moscow, 1990), pages 19–26. SIAM, Philadelphia, PA, 1991.</ref>
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| ==Dirichlet-to-Neumann operator on a bounded domain==
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| Consider a [[steady-state]] distribution of [[temperature]] in a body for given temperature values on the body surface. Then the resulting [[heat flux]] through the boundary (that is, the heat flux that would be required to maintain the
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| given surface temperature) is determined uniquely. The mapping of the surface temperature to the surface heat flux is a Poincaré–Steklov operator. This particular Poincaré–Steklov operator is called the Dirichlet to Neumann (DtN) operator. The values of the temperature on the surface is the [[Dirichlet boundary condition]] of the [[Laplace equation]], which describes the distribution of the temperature inside the body. The heat flux through the surface is the [[Neumann boundary condition]] (proportional to the [[normal derivative]] of the temperature).
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| Mathematically, for a function <math>u</math> [[harmonic function|harmonic]] in a domain <math>\Omega\subset R^n</math>, the Dirichlet-to-Neumann operator maps the values of <math>u</math> on the boundary of <math>\Omega</math> to the normal derivative <math>\partial u/\partial n</math> on the boundary of <math>\Omega</math>. This Poincaré–Steklov operator is at the foundation of [[iterative substructuring]].<ref name="Quarteroni">Alfio Quarteroni and Alberto Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, 1999</ref>
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| [[Alberto Calderon|Calderón]]'s inverse boundary problem is the problem of finding the coefficient of a divergence form elliptic partial differential equation from its Dirichlet-to-Neumann operator. This is the mathematical formulation of [[electrical impedance tomography]].
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| ==Dirichlet-to-Neumann operator for a boundary condition at infinity==
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| The solution of partial differential equation in an [[Domain (mathematical analysis)|external domain]] gives rise to a Poincaré–Steklov operator that brings the boundary condition from infinity to the boundary. One example is the Dirichlet-to-Neumann operator that maps the given temperature on the boundary of a cavity in infinite medium with zero temperature at infinity to the heat flux on the cavity boundary. Similarly, one can define the Dirichlet-to-Neumann operator on the boundary of a sphere for the solution for the [[Helmholtz equation]] in the exterior of the sphere. Approximations of this operator are at the foundation of a class of method for the modeling of acoustic scattering in infinite medium, with the scatterer enclosed in the sphere and the Poincaré–Steklov operator serving as a non-reflective (or absorbing) boundary condition.<ref name="Oberai">Assad A. Oberai, Manish Malhotra, and Peter M. Pinsky, On the implementation of the Dirichlet-to-Neumann radiation condition for iterative solution of the Helmholtz equation. Appl. Numer. Math., 27(4):443–464, 1998.</ref>
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| ==Poincaré–Steklov operator in electromagnetics==
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| The Poincaré–Steklov operator is defined to be the operator mapping the time-harmonic (that is, dependent on time as <math>e^{i\omega t}</math>) tangential electric field on the boundary of a region to the equivalent electric current on its boundary.<ref>L. F. Knockaert, On the complex symmetry of the Dirichlet-to-Neumann operator, Progress in Electromagnetics Research B, Vol. 7, 145–157, 2008. {{doi|10.2528/PIERB08022102}}</ref>
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| ==See also==
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| *[[Fluid-structure interaction]] (boundary/interface) analysis
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| *[[Schur complement method|Schur complement domain decomposition method]]
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| ==References==
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| * [[Vyacheslav Ivanovich Lebedev|Lebedev]], V. I.; Agoshkov, V. I. Operatory Puankare-Steklova i ikh prilozheniya v analize. (Russian) [Poincaré Steklov operators and their applications in analysis] Akad. Nauk SSSR, Vychisl. Tsentr, Moscow, 1983. 184 pp. {{MathSciNet|id= 87i:35053}}
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| * Vassilevski, P. S. Poincaré–Steklov operators for elliptic difference problems. C. R. Acad. Bulgare Sci. 38 (1985), no. 5, 543—546. {{MathSciNet|id= 86k:39007}}
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| <references/>
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| * E.B. Curtis, D. Ingerman, J.A. Morrow. Circular planar graphs and resistor networks. Linear Algebra and its Applications. Volume 283, Issues 1–3, 1 November 1998, Pages 115–150.
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Poincare-Steklov Operator}}
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| [[Category:Domain decomposition methods]]
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I'm Helaine (25) from All Saints South Elmham, United Kingdom.
I'm learning Bengali literature at a local university and I'm just about to graduate.
I have a part time job in a university.
http://brokerltd.com - CPM
Look at my page - publisher