Dirichlet boundary condition

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

Table of contents
- Dirichlet boundary data Dirichlet boundary condition
- Perfectly conducting thermal boundary condition Dirichlet boundary condition

Dirichlet boundary data ( $u ∣_{\partial Ω} = φ$ )

Dirichlet boundary data prescribe the boundary values of a solution as a function

φ

on the boundary of its domain. Continuity on a polygon requires agreement of the values at corners, but continuity alone does not ensure solvability if the equation loses uniform ellipticity at a boundary side.

Perfectly conducting thermal boundary condition

 0  0

In a fixed-temperature convection problem, the boundary temperature is prescribed, and its perturbation satisfies a thermal Dirichlet boundary condition. A homogeneous condition means the boundary follows the prescribed conductive background temperature. This does not necessarily mean a vertical wall has a spatially constant temperature; that additional assumption can conflict with a background vertical gradient.