Gradient descent for functionals?

Question

If $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ is smooth, then given an initial point $x_0\in\mathbb{R}^2$, we can use gradient descent to find a sequence of points $\{x_i\}_{i=1}^{\infty}$ that converges to a critical point of $f$ (if one exists). I was wondering if there is any equivalent method for functionals? That is, is there any way of generating a sequence of functions $\{g_i\}_{i=1}^{\infty}$ (preferably polynomials?) that converges to the function $g$ that minimises a functional $E[\phi]$? For example, we could take $E[\phi]$ to be the Dirichlet energy of $\phi$: $$ E[\phi] = \int_\Omega{\|\nabla\phi(x)\|^2\,\text{d}V}\,. $$ I have briefly looked at Sobolev gradients but I don't quite understand it. This is slightly different to my specific problem, as the functional I am interested is the vector Dirichlet energy: $$ E[\mathbf{v}] = \int_\Omega\left((\nabla \cdot \mathbf{v})^2 + \|\nabla \times \mathbf{v}\|^2\right)\,\text{d}V\,. $$ with conditions $$ \begin{array}{c}\langle \mathbf{v}, \hat{\mathbf{n}}\rangle=0 \mathrm{\ on\ } \partial \Omega,\\ \int_{\Omega} \|\mathbf{v}\|^2\,dV = 1.\end{array} $$ Edit: The case that I am interested specifically is when $\Omega$ is a torus with a slightly perturbed boundary or an infinite tube twisted helically around the $z$-axis.