Poincaré–Steklov operator

In

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.[1]

Dirichlet-to-Neumann operator on a bounded domain

Consider a

Mathematically, for a function ${\displaystyle u}$ harmonic in a domain ${\displaystyle \Omega \subset R^{n}}$, the Dirichlet-to-Neumann operator maps the values of ${\displaystyle u}$ on the boundary of ${\displaystyle \Omega }$ to the normal derivative ${\displaystyle \partial u/\partial n}$ on the boundary of ${\displaystyle \Omega }$. This Poincaré–Steklov operator is at the foundation of

The solution of partial differential equation in an 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 methods 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.^[3]

Poincaré–Steklov operator in electromagnetics

The Poincaré–Steklov operator is defined to be the operator mapping the time-harmonic (that is, dependent on time as ${\displaystyle e^{i\omega t}}$) tangential electric field on the boundary of a region to the equivalent electric current on its boundary.^[4]

See also

• Fluid-structure interaction
  (boundary/interface) analysis
• Schur complement domain decomposition method

References