Two-level spectral methods for nonlinear elliptic equations with multiple solutions. (English) Zbl 1398.65107
Summary: 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.
MSC:
| 65H10 | Numerical computation of solutions to systems of equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
Keywords:
nonlinear elliptic equation; multiple solutions; two-level algorithm; spectral method; polynomial system; homotopy continuationSoftware:
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