The present paper provides a two-level framework based on spectral methods and homotopy continuation for solving second-order nonlinear boundary value problems exhibiting multiple solutions. Our proposed method consists of two steps: (i) solving the nonlinear problems using low-order polynomials or a small number of collocation points, and (ii) solving the corresponding linearized problems by high-order polynomials or a large number of collocation points. The resulting two-level spectral method enjoys the following merits: (i) it guarantees multiple solutions, (ii) the computational cost is relatively small, and (iii) it is of proven high-order accuracy. These claims are supported by the detailed error estimates for semilinear equations and extensive numerical experiments of both semilinear and fully nonlinear equations.
All Science Journal Classification (ASJC) codes
- Computational Mathematics
- Applied Mathematics