We introduce a weakly over-penalized nonsymmetric interior penalty (WOPNIP) method for solving elliptic boundary value problems. The WOPNIP method introduced has h-optimal error estimates in both the energy norm and L2 norm. Also, although the resulting global stiffness matrix has a condition number of order h^(-4), there is a simple preconditioner that reduces the condition number to h^(-2). This is a significant advantage when designing a good smoothing procedure for multigrid algorithms. We then prove that there is a bound (<1) for the contraction number of the W-cycle algorithm, which is independent of the mesh level, for an appropriately chosen number of smoothing steps.