Let \(\Omega\subset \mathbb{R}^ 2\) be a bounded polygon and \(\alpha=(\alpha_ 1,\alpha_ 2)\) a unit vector. The author considers the following class of constant-coefficient convection-diffusion equations: (1) \(u_ \alpha-\sigma_ 1u_{xx}-\sigma_ 2u_{yy}=f\), where \((x,y)\in \Omega\), \(u_ \alpha=\alpha\cdot\bigtriangledown u\) and \(\sigma_ 1\) and \(\sigma_ 2\) are nonnegative. Equation (1) may be hyperbolic, parabolic or elliptic depending upon the number of nonzero diffusion coefficients which appear. The following extension of the discontinuous Galerkin method is proposed to (1): \[ \begin{multlined} (u_ \alpha^ h- \sigma_ 1u_{xx}^ h- \sigma_ 2u_{yy}^ h,v^ h)- \int_{\Gamma_{in}(T)}[(u^ h)^ +-(u^ h)^ -]v^ h\alpha\cdot n+\\ \int_{\Gamma_{in}^*(T)}\{\sigma_ 1[(u_ x^ h)^ +-(u_ x^ h)^ -]n_ 1+\sigma_ 2[(u_ y^ h)^ +- (u_ y^ h)^ - ]n_ 2\}v^ h=(f,v^ h), \quad \text{for all}\quad v^ h\in P_ n(T).\end{multlined}\tag{2} \] Here \(\Gamma_{in}(\Omega)\) is the ``inflow'' portion of \(\Gamma=\partial\Omega\) defined by \(\alpha\cdot n0\), which means that the triangle sides are to be bounded away from the characteristic direction (\(\Omega\) is divided into a quasi-uniform mesh of triangles of side length \(h\)). Two specific examples (elliptic and parabolic equations) illustrate the used method.