Consider the problem of solving for $u$ where $-\Delta u = f$, $[u] = [g]$ where $[\cdot]$ denotes cohomology class and $u, f, g$ are $p$-forms on a Riemannian manifold $M$. If $g$ instead was boundary data, this could be solved by various numerical methods. For example, one could discretize $M$ into a graph and use the finite-difference method. I am wondering if numerical analysts have studied the Laplace equation with cohomological data, or if there is a nice way to reduce the cohomological-data problem to a boundary-value problem.
To illustrate why I think that reduction to the boundary-value case should be possible, let me note that it seems to be possible if $p = 1$. In that case we lift $f, g$ to functions $\tilde f, \tilde g$ on the universal cover $\widetilde M$, and restrict to a fundamental domain $\Omega$. Then we solve the problem $-\Delta \tilde u = \tilde f$, $\tilde u|\partial \Omega = \tilde g$, so $\tilde u$ is the lift of a function $u$ with $-\Delta u = f$ and for every loop $\gamma$ in $M$, we may assume that $\gamma$ is contractible or lifts to a path $\tilde \gamma$ connecting two points on $\partial \Omega$, with $$\int_\gamma u = v(\tilde \gamma(1)) - v(\tilde \gamma(0)) = h(\tilde \gamma(1)) - h(\tilde \gamma(0)) = \int_\gamma g$$ where $dv = \tilde u$, $dh = \tilde g$. However, this approach does not work if $p \geq 2$ as we do not have a suitable analogue of universal cover (I think).
