  A Maxwell solver derived from finite element method with \mathcal{O}(N)
computing cost is developed to improve the numerical dispersion properties in
relativistic particle-in-cell (PIC) simulations. The correction of the
dispersion relation of the electromagnetic wave is achieved using the
neighboring cells via an iteration scheme without decomposing into Fourier
modes. The local nature of the communication is ideally suited to massively
parallel computer architectures. This Maxwell solver constrains the Numerical
Cherenkov instability (NCI) for the ultra-relativistic drifting pair plasma in
x direction to large wave vectors for two dimensional grid. The growth rate of
NCI is suppressed by using the low pass filtering.
