We present a systematic and efficient Chebyshev spectral method using quasi-inverse technique to directly solve the second order equation with the homogeneous Robin boundary conditions and the fourth order equation with the first and second boundary conditions. The key to the efficiency of the method is to multiply quasi-inverse matrix on both sides of discrete systems, which leads to band structure systems. We can obtain high order accuracy with less computational cost. For multi-dimensional and more complicated linear elliptic PDEs, the advantage of this methodology is obvious. Numerical results indicate that the spectral accuracy is achieved and the proposed method is very efficient for 2-D high order problems.
