Two-level spectral methods for nonlinear elliptic equations with multiple solutions^\ast

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.