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.

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.

Right-hand side

Obtained by inserting the exact solution into the latter equation.

Sample solution

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

Solution.

Comparison of h-FEM (p=1), h-FEM (p=2) and hp-FEM with anisotropic refinements

Final mesh (h-FEM, p=1, anisotropic refinements):

Final mesh.

Final mesh (h-FEM, p=2, anisotropic refinements):

Final mesh.

Final mesh (hp-FEM, h-anisotropic refinements):

Final mesh.

DOF convergence graphs:

DOF convergence graph.

CPU convergence graphs:

CPU convergence graph.

hp-FEM with iso, h-aniso and hp-aniso refinements

Final mesh (hp-FEM, isotropic refinements):

Final mesh.

Final mesh (hp-FEM, h-anisotropic refinements):

Final mesh.

Final mesh (hp-FEM, hp-anisotropic refinements):

Final mesh.

DOF convergence graphs:

DOF convergence graph.

CPU convergence graphs:

CPU convergence graph.