NIST-06 (Boundary Layer)¶

This example is a singularly perturbed problem that exhibits a boundary layer along the right and top sides of the domain.

Model problem¶

Equation solved: Convection-diffusion equation with first order terms.

$-\epsilon \nabla^{2} u + 2\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} - f = 0.$

Domain of interest: $(-1, 1)^2$ .

Boundary conditions: Dirichlet, given by exact solution.

$u(x,y) = (1 - e^{-(1 - x) / \epsilon})(1 - e^{-(1 - y) / \epsilon})cos(\pi (x + y))$

where $\epsilon$ determines the strength of the boundary layer.

Obtained by inserting the exact solution into the latter equation.

Solution for $\epsilon = 10^{-1}$ :