NIST-07 (Boundary Line Singularity)

Many papers on testing adaptive algorithms use a 1D example with a singularity of the form at the left endpoint of the domain. This can be extended to 2D by simply making the solution be constant in .

Model problem

Equation solved: Poisson equation

(1)

Domain of interest: Unit Square

Boundary conditions: Dirichlet, given by exact solution.

Exact solution

where determines the strength of the singularity.

Right-hand side

Obtained by inserting the exact solution into the equation.

Sample solution

Solution for :

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

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

DOF convergence graphs:

CPU convergence graphs:

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

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

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

DOF convergence graphs:

CPU convergence graphs: