NIST-10 (Interior Line Singularity)¶

This example is an extension of Boundary Line Singularity (NIST-07) with an anisotropic solution to allow for a sloped line so that the singularity does not necessarily coincide with element edges.

Model problem¶

Equation solved: Poisson equation

(1) $-\Delta u - f = 0.$

Domain of interest: $(-1, 1)^2$ .

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

$u(x,y) = \cos(Ky)\ \ \ \mbox{for}\ x \le 0,\\ u(x,y) = \cos(Ky) + x^{\alpha}\ \ \ \mbox{for}\ x > 0,$

where $K$ and $\alpha$ are real constants.

Right-hand side¶

Obtained by inserting the exact solution into the equation.

Sample solution¶

Solution for $K = \pi/2$ and $\alpha = 2.01$ :

NIST-10 (Interior Line Singularity)¶

Model problem¶

Exact solution¶

Right-hand side¶

Sample solution¶

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NIST-10 (Interior Line Singularity)¶

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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