AbstractThe aim of this work is to give theoretical justification of several types of finite element approximations to the initial‐boundary value problems of first order linear hyperbolic equations. Our approximate scheme is obtained by the piecewise linear continuous finite element method for space variable,x, and the Euler type step by step integration method for time variable,t. An artificial viscosity technique, up‐stream type methods are considered within the frame work ofL2‐theory. The convergence and the error estimate of the approximate solutions to the true one are disc