On the numerical solution of double-periodic elliptic eigenvalue problems (Q1262101)

The computation of solutions \(u=u(x,\lambda)\) of the eigenvalue problem \(\Delta u+\lambda (u+f(u))=0,\) \(\lambda >0,\) is considered which are \(2\pi\)-periodic in x and \(2\pi\) /\(\alpha\)-periodic in \(\lambda\) with prescribed \(\alpha\), notable \(\alpha =3^{1/2}.\) Subspaces V with the desired symmetry properties are constructed and a Ritz method is applied for the discretization in V. The resulting finite-dimensional bifurcation problem is solved by an algorithm of \textit{H. B. Keller}, \textit{W. F. Langford} [Arch. Rat. Mech. Analysis 48, 83-158 (1972; Zbl 0249.47058)]. If \(f: R\to R,\) \(f(0)=f'(0)=0,\) is entire or a polynomial and V an algebra, then the Ritz approximation is stable with respect to perturbations with reduced symmetry. Some examples illustrate the procedure.