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 $\alpha = \frac{1}{10 \pi}$ :

NIST-08 (Oscillatory)¶

Model problem¶

Exact solution¶

Right-hand side¶

Sample solution¶

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NIST-08 (Oscillatory)¶

Model problem¶

Exact solution¶

Right-hand side¶

Sample solution¶

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

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

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