On the numerical solution of double-periodic elliptic eigenvalue problems

In the present paper, numerical solving of the double-periodic elliptic eigenvalue problems $$M(u,\lambda ): = \Delta u + \lambda (u + f(u)) = 0, 0 \leqslant x< 2\pi ,0 \leqslant y< 2\pi /\sqrt {3,} $$ is considered regarding special symmetry properties. At first, subspacesV with the desired symmetry are constructed then a classical Ritz method is applied for the discretization inV and the resulting finite-dimensional bifurcation problem is solved by an algorithm proposed by Keller and Langford representing anumerical implementation of the Ljapunov-Schmidt procedure. Iff(u) is an entire function or a polynomial andV is an algebra then the computed solutions reveal to be stable with respect to perturbations of less symmetry. Some examples demonstrate the efficiency of the procedure.