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}.

Boundary conditions

Nonconstant Dirichlet, matching the exact solution.

Sample solution


Convergence comparison

Final mesh (h-FEM with linear elements):

Final mesh (h-FEM with linear elements).

Final mesh (h-FEM with quadratic elements):

Final mesh (h-FEM with quadratic elements).

Final mesh (hp-FEM):

Final mesh (hp-FEM).

DOF convergence graphs:

DOF convergence graph.

CPU time convergence graphs:

CPU convergence graph.

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