Solving scattering problems using the stabilized finite element and the algebraic multigrid

Electromagnetic waves are waves that propagate without the need of medium. One of the equations to describe the phenomenon of electromagnetic wave scattering problems is the Helmholtz equation. There are difficulties in solving the Helmholtz equation numerically, two of which that the solution form is highly oscillatory and the numerical solution of the high-frequency cases has low accuracy, which is called "pollution effects". Due to the limitation of memory and CPU size in a digital computer, simulating this problem with a large size of computational points is impossible. Besides, solving the linear system of the Helmholtz equation is quite challenging. It is difficult to find an efficient iterative solver that can converge with only a few iterations. This research presents a numerical scheme to minimize the "pollution effect"that appear in the numerical solution of the Helmholtz equation. The numerical scheme gives better accuracy than the standard scheme of the finite element. Moreover, we solve the linear system by using an algebraic multigrid as a preconditioner in a main iteration routine, which is based on a preconditioned variant of generalized conjugate residual (GCR) methods. This solver can give a better performance regarding its number of iteration and the time cost.