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:

Computational domain.

Exact solution

(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).

Boundary conditions

Nonconstant Dirichlet, matching the exact solution.

Sample solution


Convergence comparisons

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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