  We derive several $L^\infty$ error estimates for the symmetric interior
penalty (SIP) discontinuous Galerkin (DG) method applied to the Poisson
equation in a two-dimensional polygonal domain. Both local and global estimates
are examined. The weak maximum principle (WMP) for the discrete harmonic
function is also established. We prove our $L^\infty$ estimates using this WMP
and several $W^{2,p}$ and $W^{1,1}$ estimates for the Poisson equation.
Numerical examples to validate our results are also presented.
