NIST-08 (Oscillatory)

This problem is inspired by the wave function that satisfies a Shrodinger equation model of two interacting atoms. It is highly oscillatory near the origin, with the wavelength decreasing closer to the origin.

Model problem

Equation solved: Hemholtz equation

-\nabla^{2} u - \frac{1}{(\alpha + r)^{4}} u - f = 0.

where r = \sqrt{x^{2} + y^{2}}. The number of oscillations, N, is determined by the parameter \alpha = \frac{1}{N \pi}

Domain of interest: Unit Square (0, 1)^2.

Boundary conditions: Dirichlet, given by exact solution.

Exact solution

u(x,y) = sin(\frac{1}{\alpha + r})

Right-hand side

Obtained by inserting the exact solution into the equation.

Sample solution

Solution for \alpha = \frac{1}{10 \pi}:

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.