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 to the values of another boundary condition.
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.
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.
The heat flux through the surface is the Neumann boundary condition (proportional to the normal derivative of the temperature).
[2] 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.
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.
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.