In this post, I’ll introduce the FE formulation of a generalized linear and coupled weak form. Said weak formulation has the form

Find $u\in V_1$, $y\in V_2$ such that

for all $v\in V_1$, $w \in V_2$ where $a(\cdot, \cdot): V_1\times V_1 \to \IR$, $b(\cdot, \cdot): V_2\times V_1 \to \IR$, $d(\cdot, \cdot): V_1\times V_2 \to \IR$, $e(\cdot, \cdot): V_2\times V_2 \to \IR$ are bilinear forms and $c(\cdot): V_1\to \IR$, $f(\cdot): V_2\to \IR$ are linear forms.

Here, the objective is to solve for the two unknown functions $u$ and $y$. One can also imagine an arbitrary degree of coupling between $n$ variables with $n$ equations.

We introduce the following discretizations

where the corresponding number of shape functions are $n_n^1$ and $n_n^2$, respectively.

Substituting the discretizations in \eqref{eq:coupledweakform1}, we obtain two linear systems of equations

for $I=1,\dots,n_n^1$ and $K=1,\dots,n_n^2$.

We write this system as

where the components of given matrices and vectors are defined as

Solution of \eqref{eq:coupledsystem1} yields the unknown vectors $\Bu$ and $\By$.