Two explicit finite element Galerkin methods are considered for first-order hyperbolic equations of the form \(\alpha\cdot \nabla u+ \beta u= f\) in \(V\) (\(V\) a bounded polygonal domain in \(\mathbb{R}^2\)), where \(u= g\) on a boundary \(dV\), which is described as an inflow boundary and is defined so that \(\alpha\cdot n(x)< 0\), where \(n\) is the outward normal at \(x\). One of these methods uses a piecewise polynomial approximation and the other uses a discontinuous approximation. The performance of these methods is considered and, for the discontinuous method, a new error estimate is given for the piecewise constants. Numerical examples are given to illustrate the theory.