The paper is devoted to Galerkin solutions of the Wiener-Hopf equation \[ x(t)- \int^\infty_0 h(t-\tau) x(\tau)d\tau= f(t),\quad t\in[0,\infty),\tag{1} \] where the right-hand side \(f\) and the kernel function \(h(t)\) are given, \(x:[0,\infty)\to\mathbb{R}\) is the unknown solution and \(h(t)\) is smooth. It is assumed that there exists a \(\mu^*>0\) such that for \(l\geq 0\), \[ \int^\infty_{-\infty} e^{\mu^*|t|}|D^lh(t)|dt0.\end{cases} \] The approximate solution of equation (1) is chosen as piecewise polynomial function. The unknown coefficients of polynomials are determined with the Galerkin method. The unique solution of the Galerkin approximation is proved. Then a kind of iterative correction method for the Galerkin approximation is offered and it is shown that this is not only a high-order method but also an adaptable one.