Screen (Maxwell’s Equations)

Git reference: Benchmark screen.

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:

Boundary conditions

Tangential component of solution taken from known exact solution (essential BC). See the main.cpp file.

Exact solution

This is rather complicated in this case - please look into the corresponding file exact_solution.cpp.

Weak forms

template<typename Real, typename Scalar>
Scalar bilinear_form(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u, Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext)
{
  return int_curl_e_curl_f<Real, Scalar>(n, wt, u, v) - int_e_f<Real, Scalar>(n, wt, u, v);
}

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):

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 (hp-FEM):

DOF convergence graphs:

CPU time convergence graphs: