Git reference: Benchmark 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.
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.
u(x,y) = sin(\frac{1}{\alpha + r})
Obtained by inserting the exact solution into the equation.
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:
Final mesh (hp-FEM, isotropic refinements):
Final mesh (hp-FEM, h-anisotropic refinements):
Final mesh (hp-FEM, hp-anisotropic refinements):
DOF convergence graphs:
CPU convergence graphs: