Interior Layer (Elliptic)¶

This example has a smooth solution that exhibits a steep interior layer.

Model problem¶

Equation solved: Poisson equation

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

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

Exact solution¶

(2) $u(x, y) = \mbox{atan}\left(S \sqrt{(x-1.25)^2 + (y+0.25)^2} - \pi/3\right).$

where $S$ is a parameter (slope of the layer). With larger $S$ , this problem becomes difficult for adaptive algorithms, and at the same time the advantage of adaptive $hp$ -FEM over adaptive low-order FEM becomes more significant. We will use $S = 60$ in the following.

Right-hand side¶

Obtained by inserting the exact solution into the equation:

(3) $f(x, y) = \frac{27}{2} (2y + 0.5)^2 (\pi - 3t) \frac{S^3}{u^2 t_2} + \frac{27}{2} (2x - 2.5)^2 (\pi - 3t) \frac{S^3}{u^2 t_2} - \frac{9}{4} (2y + 0.5)^2 \frac{S}{u t^3} - \frac{9}{4} (2x - 2.5)^2 \frac{S}{u t^3} + 18 \frac{S}{ut}.$