I need help with the finite element method for the following problem I present in weak formulation. Certain details are left out since they are not important for the essence of this question. This is actually an eigenproblem for the model of the curved rod in mathematical elasticity.
The question is, generally, how does one construct the finite-dimensional subspace when the objective is to approximate vector-valued functions? In particular, consider the following problem.
Let $V=\{(v,w)\in H^1(0,l)^3\times H^1(0,l)^3: v'+t\times w=0, v(0)=w(0)=\mathbf{0}\}$. Here, $l>0$, $H^1(0,l)^3$ is the Sobolev $H^1$ space of functions $(0,l)\mapsto\mathbb{R}^3$ and $t$ is also some function with values in $\mathbb{R}^3$.
We want to obtain all $\lambda\in\mathbb{R}$ which for there exist $(u,\omega)\neq\mathbf{0}\in V$ such that: \begin{equation} \tag{1} \label{problem_statement} \int_{0}^{l}Q(x)HQ(x)^T\omega(x)'\cdot w(x)'=\lambda\int_{0}^{l}\rho(x)A(x)u\cdot v, \hspace{0.2cm}\forall(v,w)\in V\ \end{equation} $Q(x)$ is an orthonormal matrix for any $x\in(0,l)$, $H$ can be considered a constant real symmetric $3\times 3$ matrix while $\rho, A\in\mathcal{C}(0,l; \mathbb{R})$.
I ought to use FEM to numerically solve this problem and I cannot seem to figure out how to do this. After we define two bilinear forms $B,L:V\times V\to\mathbb{R}$: \begin{align*} B((u,\omega),(v,w))&=\int_{0}^{l}QHQ^T\omega'\cdot w'\\ L((u,\omega),(v,w))&=\int_{0}^{l}\rho A u\cdot v \end{align*} $(1)$ becomes: \begin{align*} B((u,\omega),(v,w))=\lambda L((u,\omega),(v,w)) \end{align*}
Now I cannot figure out how to define the finite-dimensional subspace for FEM. Let us denote with $\widetilde{V_n}$ the usually used space of Lagrange's first degree polynomials from $(0,l)$ to $\mathbb{R}$. I tried setting $V_n=\widetilde{V_n}^3$. For instance, the function $u$ approximation $u_n\in V_n$ is thus given with component-wise approximation: \begin{align*} u_n(x)=\sum_{i=1}^{n}(\alpha_i^1\phi_i(x),\alpha_i^2\phi_i(x),\alpha_i^3\phi_i(x)) \end{align*} Here $\phi_i\in V_n$ and $\alpha_i^j\in\mathbb{R}$ for $j=1,2,3, i=1,2,\dots,n$.
This, unfortunately, has not lead me anywhere. I know my question might resemble this one: System of equations for vector valued functions problems. However, I obviously tried the approach suggested there.
Any help is immensely appreciated.