Starting with an example:
I want to apply a numerical method (e.g. the finite difference method) to find a solution of a partial differential equation
$$ \nabla^2 f(\mathbf{x}) = k f(\mathbf{x}) $$
where $\mathbf{x} \in R^D$, $\nabla^2 = \sum_{d=1}^{D} \frac{\partial^2}{\partial x_d ^2}$ and $D >> 3$.
Applying an ordinary finite difference scheme would result in the exponentially-growing number of grid points.
Is there any approximate numerical method that can handle such exponential increase of computation in higher dimension?
Moreover, how can boundary conditions be specified in such a approximate method?