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In this paper, we consider the following nonlinear wave equations: $$\begin{array}{l} \frac{{\partial ^2 \phi }}{{\partial t^2 }} - \frac{{\partial ^2 \phi }}{{\partial x^2 }} + \mu ^2 \phi + v^2 \chi ^2 \phi + f\left( {|\phi |^2 } \right)\phi = 0, \\ \frac{{\partial ^2 \phi }}{{\partial t^2 }} - \frac{{\partial ^2 \chi }}{{\partial x^2 }} + \alpha ^2 \chi + v^2 \chi |\phi |^2 + g(\chi ) = 0 \\ \\ \end{array}$$ with the periodic-initial conditions: $$\begin{array}{l} \phi (x---\pi ,t) = \phi (x + \pi ,t), \chi (x---\pi ,t) = \chi (x + \pi ,t), \\ \phi (x,0) = \hat \phi _0 (x), \phi _t (x,0) = \hat \phi _1 (x), \\ \chi (x,0) = \hat \chi _0 (x), \chi _t (x,0) = \hat \chi _1 (x), \\ - \infty< x< \infty , 0 \le t \le T. \\ \end{array}$$

Abstract: In this paper, we shall give two new variational formulations for Wilson element on arbitrary quadrilateral meshes. It is shown that the convergence order isO(h) as the Wilson element with original variational formulation. Furthermore, the multigrid method of the Wilson element with the two new variational formulations is developed, and the order of convergence is alsoO(h).