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.