We show how to construct higher-order (10 and above) finite-difference schemes using Minimum Sobolev Norm techniques that lead to O(N2) condition number discretizations of second-order elliptic PDEs. Numerical experiments comparing with standard FEM solvers show the benefit of the new approach. We then discuss fast numerical methods to convert the discrete equations into discretized integral equations that can have bounded condition number, and hence achieve even higher numerical accuracy.