The rate of convergence of the finite element method for the Dirichlet problem for the \(p(x)\)-Laplacian \((1\leq p_1\leq p(x)\leq p_2\leq 2)\) in a two-dimensional convex domain \(\Omega\) with Lipschitz boundary is studied. The paper is organized as follows. Section 1 is an introduction. In Section 2, some preliminary facts are given. The main results are proved in Section 3. A family of numerical examples is given in Section 4. When the decomposition-coordination method for approximating the solution is used, the behavior of the error is studied.