L-Shape (Elliptic)¶

This is a standard adaptivity benchmark whose exact solution is smooth but contains singular gradient in a re-entrant corner.

Model problem¶

Equation solved: Laplace equation

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

Domain of interest:

(2) $u(x, y) = r^{2/3}\sin(2a/3 + \pi/3)$

where $r(x,y) = \sqrt{x^2 + y^2}$ and $a(x,y) = \mbox{atan}(x/y)$ .

Nonconstant Dirichlet, matching the exact solution.

Final mesh (h-FEM with linear elements):

Final mesh (h-FEM with quadratic elements):

Final mesh (hp-FEM):

DOF convergence graphs:

CPU time convergence graphs: