NIST-12 (Multiple Difficulties)¶

This problem combines four difficulties of different strengths into the same problem by combining some of the features of the other test problems (reentrant corner, sharp peak, etc).

Model problem¶

Equation solved: Poisson equation

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

Domain of interest: L-shaped domain $(-1,1) \times (-1,1)$ \ $(0,1) \times (-1,0)$ .

Boundary conditions: Dirichlet, given by exact solution.

Right-hand side¶

Quite complicated, see the source code.

Exact solution¶

$u(x,y) = r^{\alpha_{C} }\sin(\alpha_{C} \theta) + e^{-\alpha_{P} ((x - x_{P})^{2} + (y - y_{P})^{2})} + tan^{-1}(\alpha_{W} (r_{W} - r_{0})) + e^{-(1 - y) / \epsilon}.$

where $\alpha_C = \pi / \omega_C$ , $r = \sqrt{x^2+y^2}$ and $\theta = tan^{-1}(y/x)$ . Here $\omega_C$ determines the angle of the re-entrant corner.

$(x_{P}, y_{P})$ is the location of the peak, $\alpha$ determines the strength of the peak,

and $r_{W} = \sqrt{(x - x_{W})^{2} + (y - y_{W})^{2}}$ . Here $(x_{W}, y_{W})$ is the center of the circular wave front. $r_{0}$ is the distance from the wave front to the center of the circle, and $\alpha_W$ gives the steepness of the wave front.

Last but not least, $\epsilon$ determines the strength of the boundary layer; the boundary layer was placed at $y = -1$ .

Sample solution¶

Solution for $\omega_C = 3 \pi /2$ , $(x_{W}, y_{W}) = (0, -3/4)$ , $r_{0} = 3/4$ , $\alpha_{W} = 200$ , $(x_{P}, y_{P}) = (\sqrt{5} / 4, -1/4)$ , $\epsilon = 1/100$ :