Weak form is an important tool for the analysis of electromagnetic problems. It is a bridge from the partial differential equations to the linear algebra. For example, the Helmholtz equation with wave number \(k_c\) is
\(\nabla_t^2 \phi+k_c^2 \phi=0\)
It is so-called “strong form”, in which the unknow is in the 2nd order differential operator. It is stuiable for the finite difference method. However it is not stuiable for the finite element method. Somebody invented the weak form of the parital differential equations by multiplying both sides with a testing function T and integrating over the surface F. That is
\(\iint[T(\nabla_t^2 \phi+k_c^2 \phi]ds=0\)
or
\(\iint[T(\nabla_t^2 \phi ds=-k_c^2\iint T \phi]ds=0\)
Use the vector identity \(\nabla(f A)=A\cdot \nabla f+f\nabla\cdot A\) to simplify the first term, let \(f=T, A=\nabla \phi\)
\(\iint \nabla \cdot[T(\nabla \phi)]ds=\iint \nabla T \cdot \nabla \phi ds+\iint T(\nabla\cdot\nabla\phi)ds\)
and use the divergency theorem to simplify the left side
\(\iint \nabla\cdot[T(\nabla\phi)]ds=\int\nabla\phi\cdot \hat n dl=\int T \frac{\partial\phi}{\partial n} dl\)
Where \(\hat n\) is the unit normal vector along the boundary \(dF\). So,
\(\iint(\nabla T\cdot\nabla \phi)ds-k_c^2 \iint T \phi ds=\int T \frac{\partial\phi}{\partial n}dl\)
If we set \(T=\phi\), T vanishes on the PEC boundary for TM mode. \(\frac{\partial\phi}{\partial n}\) vanishes for the TE mode. So the weak form of the helmholtz equation inside the waveguide is
\(\iint\nabla (T \cdot \nabla \phi )ds=k_c^2\iint T\phi ds\)