The Finite Element Method in Electromagnetics, 3rd Edition

APPENDIX BTHE RITZ PROCEDURE FOR COMPLEX-VALUED PROBLEMS

In Chapter 2, whether in the formulation of the classical Ritz solution or in the finite element solution, we assumed a real-valued problem for the sake of simplicity. This restriction can, however, be lifted and to illustrate this, let us consider the classical Ritz procedure. The treatment in the finite element method is very similar.

For a complex-valued problem, the unknown function ϕ is complex. The functional given in (2.5) can be written as

(B.1)

The trial function can be expressed as a combination of some expansion functions

(B.2)

which is the same as (2.6), but now c_j are the complex expansion coefficients. However, the expansion or basis functions v_j are always chosen to be real.

Substituting (B.2) into (B.1), we obtain

(B.3)

Because ∂c*_i/∂c_i has no unique value, in this case we cannot take the partial derivative of F with respect to c_i as is done for the real-valued problem. Instead, we should take the partial derivative of F with respect to the real and imaginary parts of c_i, denoted as c^r_i and cⁱ_i, respectively. Let us take the partial derivative of F with respect to c^r_i first, giving

(B.4)

As a result ...