Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859).[1] When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain.

In Finite Element Method, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.[2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary condition.

The question of finding solutions to such equations is known as the Dirichlet problem. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition.

where 2 denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rn take the form

For example, the following would be considered Dirichlet boundary conditions:

Many other boundary conditions are possible, including the Cauchy boundary condition and the mixed boundary condition. The latter is a combination of the Dirichlet and Neumann conditions.