Hybrid IPDGFEM/FDM method for Maxwell's equations
In this example we present numerical simulation of the hybrid Interior Penalty Discontinuous Galerkin Finite Element Method/ Finite Difference Method (IPDGFEM/FDM) for the second order time-dependent Maxwell's equations for the electric field. First we consider the case of constant coefficients in 2D.
In this particular case solutions (E_1, E_2) of the electric field should correspond to the solution of the obvious wave equation.
In the test below we show comparison of the exact and computed solutions for the electric field for hybrid IPDGFEM/FDM method in this particular case.
In the case when exact components of the electric field (E_1, E_2) at every point of the computational domain with coordinates (x,y) at the time moment t are prescribed as below
E1(x,y) = ((t*t)/2.0)*cos(M_PI*x)*sin(M_PI*y), (1)
E2(x,y) = -((t*t)/2.0)*sin(M_PI*x)*cos(M_PI*y), (2)
the right hand side in the time-dependent Maxwell's equation for the electric field will be defined as
F_1(x,y) = cos(M_PI*x)*sin(M_PI*y)*(1.0 + t*t*M_PI*M_PI);
F_2(x,y) = sin(M_PI*x)*cos(M_PI*y)*(-1.0- t*t*M_PI*M_PI);
So, we have to compute hybrid method for above prescribed right hand sides (F_1, F_2) and compare it with the exact solution (E_1, E_2) given in (1)-(2).
Results of computations in hybrid IPDGFEM/FDM method are presented below: