Screen (Maxwell’s Equations)

This example solves time-harmonic Maxwell’s equations. It describes an electromagnetic wave that hits a thin screen under the angle of 45 degrees, causing a singularity at the tip of the screen. The strength of the singularity makes this example rather difficult.

Model problem

Equation solved: Time-harmonic Maxwell’s equations

(1)\frac{1}{\mu_r} \nabla \times \nabla \times E - \kappa^2 \epsilon_r E = \Phi.

Domain of interest is the square (-1,1)^2 missing the edge that connects the center with the midpoint of the left side. It is filled with air:

Computational domain.

Boundary conditions

Tangential component of solution taken from known exact solution (essential BC).

Exact solution

This is rather complicated in this case - see the file definitions.cpp.

Sample solution

Real part of E_1:


Real part of E_2:


Imaginary part of E_1:


Imaginary part of E_2:


Convergence comparisons

Final mesh (h-FEM with linear elements):

Final mesh (h-FEM with linear elements).

Note that the polynomial order indicated corresponds to the tangential components of approximation on element interfaces, not to polynomial degrees inside the elements (those are one higher).

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.