Poincaré–Steklov operator

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. 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.

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.

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 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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