B.-W. Jeng

A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems

We present a two-grid finite element discretization scheme with a two-loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems. First we use the predictor-corrector continuation method to compute an approximating point for the solution curve on the coarse grid. Then we use this approximating point as a predicted point for the solution curve on the fine grid. In the corrector step we solve the first and the second order approximations of the nonlinear PDE to obtain corrections for the state variable on the fine grid and the coarse grid, respectively. The continuation parameter is updated by computing the Rayleigh quotient on the fine space. To guarantee the approximating point we just obtained lies on the solution curve, we perform Newtons method. We repeat the process described above until the solution curve on the fine space is obtained. We show how the singular points, such as folds and bifurcation points, can be well approximated. Comprehensive numerical experiments show that the two-grid finite element discretization scheme with a two-loop continuation algorithm is efficient and robust for solving second order semilinear elliptic eigenvalue problems.

Liapunov-Schmidt Reduction and Continuation for Nonlinear Schro¨dinger Equations

We study the bifurcation scenario of nonlinear Schro¨dinger equations (NLS). The Liapunov-Schmidt reduction is applied to show that the simple bifurcations of a single NLS are pitchfork. The pitchfork bifurcation can be subcritical or supercritical, depending on the coefficient of the cubic term we choose. We also describe numerical methods so that the Liapunov-Schmidt reduction can effectively handle a corank-2 bifurcation point. Next, we apply numerical continuation methods to trace solution curves and surfaces of the NLS, where the system is discretized by the centered difference approximations. Numerical results on two- and three-dimensional

TRACING THE SOLUTION SURFACE WITH FOLDS OF A TWO-PARAMETER SYSTEM

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a two-grid finite element discretization scheme for nonlinear eigenvalue problems

-coupled NLS are reported, where the physical properties such as the effect of trapping potentials, isotropic and nonisotropic trapping potentials, mass conservation constraints, and strong and weak repulsive interactions are considered in our numerical experiments.

A time-independent approach for computing wave functions of the Schrödinger–Poisson system

We describe a special Gauss–Newton method for tracing solution manifolds with singularities of multiparameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each solution curve can be easily detected. Next, we choose an interval for the second continuation parameter, and trace one-dimensional solution curves for certain values in this interval. This constitutes a two-dimensional solution surface. The procedure can be generalized to trace a k-dimensional solution manifold. Numerical results in 1D, 2D and 3D second-order semilinear elliptic eigenvalue problems given by Lions [1982] are reported.

SUPERCONVERGENCE OF FEMS AND NUMERICAL CONTINUATION FOR PARAMETER-DEPENDENT PROBLEMS WITH FOLDS

We describe a two-grid finite element discretization scheme which can be used to trace solution branches as well as to detect bifurcation points of certain sec- ond order semilinear elliptic eigenvalue problems. Sample numerical results are reported.

SYMMETRY REDUCTIONS AND A POSTERIORI FINITE ELEMENT ERROR ESTIMATORS FOR BIFURCATION PROBLEMS

We describe a two-grid finite element discretization scheme for computing wave functions of the Schrodinger–Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrodinger–Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h4). Copyright

A continuation–domain decomposition algorithm for bifurcation problems

We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poissons equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.

Superconvergence of bi-k-Lagrange elements for eigenvalue problems ☆

We discuss efficient continuation algorithms for solving nonlinear eigenvalue problems. First, we exploit the idea of symmetry reductions and discretize the problem on a symmetry cell by the finite element method. Then we incorporate the multigrid V-cycle scheme in the context of continuation method to trace solution branches of the discrete problems, where the preconditioned Lanczos method is used as the relaxation scheme. Next, we apply the symmetry reduction technique to the two-grid finite element discretization scheme [Chien & Jeng, 2005] to solve some nonlinear eigenvalue problems in physical science. The two-grid centered difference discretization scheme described therein was also implemented for comparison. Sample numerical results are reported.

IMPLEMENTING MINRES AND SYMMLQ FOR EIGENVALUE PROBLEMS

We study numerical solution branches of certain parameter-dependent problems defined on compact domains with various boundary conditions. The finite differences combined with the domain decomposition method are exploited to discretize the partial differential equations. We propose efficient numerical algorithms for solving the associated linear systems and for the detection of bifurcation points. Sample numerical results are reported.