For the 2D Poisson equation, there exist on finite difference mesh, some code taking $O(n \log(n))$ operations to solve it on a mesh with $n$ nodes. They rely on Fast Fourier Transform or Block Cyclic Reduction.
Is there anything as fast on unstructured meshes? I am afraid no, but fast solution of Poisson equation on unstructured meshes is a common problem. Any known efficient solution method?