We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0, & x \in R, y \in R, 0 < z \leq 1,\\u(x,y,0) = g(x,y), & x \in R, y \in R,\\u_z(x,y,0) = 0, & x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 < z < 1$. The problem is ill-posed, the solution (if it exists) doesnâ€™t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.