On high-order finite element solution of eigenvalue problems on isospectral surfaces

Isospectral surfaces provide a rich family of benchmark problems. In this paper the efficacy of a hp-finite element method in such Laplace-Beltrami eigenvalue problems has been studied. In addition, for the p-version natural auxiliary space error estimators have been shown to be effective also in this context. As part of the numerical experiments further numerical evidence of the validity of the so-called Quarter Sphere Conjecture has been produced. For the isospectral surface pairs constructed via transplantation, the extension for perforated surfaces is derived. The lack of such constraints for surfaces with isothermal coordinates is also demonstrated.