Dirichlet boundary condition

In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. 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.

Examples

ODE

For an ordinary differential equation, for instance, $${\displaystyle y^{″}+y=0,}$$ the Dirichlet boundary conditions on the interval [a,b] take the form $${\displaystyle y(a)=\alpha ,y(b)=\beta ,}$$ where α and β are given numbers.

PDE

For a partial differential equation, for example, $${\displaystyle {\mathrm{\nabla }}^{2}y+y=0,}$$ where ${\displaystyle {\mathrm{\nabla }}^2}$ denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rⁿ take the form $${\displaystyle y(x)=f(x)\mathrm{\forall }x\in \mathrm{\partial }\mathrm{\Omega },}$$ where f is a known function defined on the boundary ∂Ω.

Applications

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

• In mechanical engineering and civil engineering (beam theory), where one end of a beam is held at a fixed position in space.
• In heat transfer, where a surface is held at a fixed temperature.
• In electrostatics, where a node of a circuit is held at a fixed voltage.
• In fluid dynamics, the no-slip condition for viscous fluids states that at a solid boundary, the fluid will have zero velocity relative to the boundary.