Band structure calculation of frequency dependent photonic crystals has important applications. The associated eigenvalue problem is nonlinear and the development convergent numerical methods challenging. In this paper, we formulate band as a holomorphic Fredholm operator function index zero. Lagrange finite elements are used to discretize operators. convergence eigenvalues proved using abstrac...

A set of linear and nonlinear stability analysis tools have been developed to analyze steady state incompressible flows in 3D geometries. The algorithms have been implemented to be scalable to hundreds of parallel processors. The linear stability of steady state flows are determined by calculating the rightmost eigenvalues of the associated generalize eigenvalue problem. Nonlinear stability is ...

We consider a nonlinear Schrödinger equation with a bounded local potential in R3. The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H1, and that its ground state component is larger than n3−ǫ with ǫ > 0 small. We prove that the solution will converge ...

a r t i c l e i n f o a b s t r a c t In this paper we use a numerical relaxation algorithm to improve and generalize the obtainment of the perturbation eigenstates of nonlinear systems. As a model problem we consider the linear stability analysis of the vortex eigenstates of the cubic–quintic nonlinear Schrödinger equation. It is shown by numerical calculations that the relaxation algorithm pe...

We introduce a new method for the calculation of bounds on the eigenvalues of Hessian matrices of twice continuously differentiable functions. Eigenvalue bounds of Hessian matrices arise in a number of notoriously difficult tasks in computational chemical engineering. For example, Hessian matrix eigenvalue bounds are used in global nonlinear optimization, global convexity/concavity analysis in ...

Abstract We study the validity of comparison and maximum principles their relation with principal eigenvalues, for a class degenerate nonlinear operators that are extremal among one-dimensional fractional diffusion.

Algorithms for nonlinear eigenvalue problems (EP), often require solving selfconsistently a large number of EP. Convergence di culties may occur if the solution is not sought in a right neighborhood; if global constraints have to be satis ed; and if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP obtained from discretizations of partial di ere...