I am trying to figure out if my weak form formulation (down to bilinear and linear form) is correct.
The strong form and BCs are as followed:
$ \nabla\cdot(\nabla u ) = k(u-v) \quad \mathrm{in} \quad \Omega $
$ \nabla\cdot(\nabla v ) = -k(u-v) \quad \mathrm{in} \quad \Omega $
$ -\nabla u = m(u-u^\prime) \quad \mathrm{on} \quad \partial\Omega $
$ -\nabla v = n(v-v^\prime) \quad \mathrm{on} \quad \partial\Omega $
where, $m,n,k$ are positive coefficients and $u^\prime, v^\prime$ are prescribed values.
With test functions $u_t, v_t$, I worked out the bilinear form $a$ to be:
$ a(u,v,u_t, v_t) = \int_\Omega \nabla u : \nabla u_t d\Omega + \int_\Omega \nabla v : \nabla v_t d\Omega + \int_\Omega k(u-v)u_t d\Omega - \int_\Omega k(u-v)v_t d\Omega + \int_{\partial\Omega} m\ u\ u_t d\partial\Omega + \int_{\partial\Omega} n\ v\ v_t d\partial\Omega $
and the linear form $\ell$ to be:
$ \ell(u_t, v_t) = \int_{\partial\Omega} m\ u^\prime\ u_t d\partial\Omega + \int_{\partial\Omega} n\ v^\prime\ v_t d\partial\Omega $
And the finite element problem is solved with $a(u,v,u_t, v_t) = \ell(u_t, v_t)$
I must made some mistakes somewhere and unable to see it, could anyone help pointing them out? I really appreciate this!
