Is there good approximation for largest (in magnitude) eigenvalue for discrete Laplacian ($\nabla^2$) obtained from nonuniform structured grid (like that)?Of course, one can always use general methods such as Lanczos or Power Iteration, but I am interested in cheaper ways, possibly less accurate.I found exact expressions for eigenvalues and eigenvectors of Laplacian in case of an uniform grid in Finite Difference Methods for Ordinary and Partial Differential Equations, p.21Obviously these formulas can't be extended for nonuniform grid case.I've tried Gerschgorin disks theorem. Downloaded this matlab script and created 5-point 2D Laplacian for uniform grid using following lines:m=10;h=10;I = speye(m);e = ones(m,1);T = spdiags([e -4*e e],[-1 0 1],m,m);S = spdiags([e e],[-1 1],m,m);A = (kron(I,T) + kron(S,I)) / h^2;Result looks like that:Bounds well enough. 
Largest eigenvalue of FD discrete Laplacian
eigensystem;finite difference