A two-grid discretization scheme for semilinear elliptic eigenvalue problems. (English) Zbl 1095.65100

Chien, C.-S.; Jeng, B.-W.

SIAM J. Sci. Comput. 27, No. 4, 1287-1304 (2006).

Summary: 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 partial differential equation 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 Newton’s 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.

MSC:

65N25	Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65P30	Numerical bifurcation problems
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

two-grid scheme; finite element method; singular points; semilinear elliptic eigenvalue problems; predictor-corrector continuation method; Newton’s method; bifurcation points; numerical experiments