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

Yingwei Wang, Wenrui Hao, Guang Lin

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)B1180-B1205
JournalSIAM Journal on Scientific Computing
Volume40
Issue number4
DOIs
StatePublished - Jan 1 2018

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Two-level Method
Nonlinear Elliptic Equations
Multiple Solutions
Spectral Methods
Collocation
Homotopy Continuation
Polynomials
Fully Nonlinear Equations
High Order Accuracy
Polynomial
Semilinear Equations
Nonlinear Boundary Value Problems
Nonlinear equations
Semilinear
Boundary value problems
Nonlinear Problem
Computational Cost
Error Estimates
Numerical Experiment
Higher Order

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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Two-level spectral methods for nonlinear elliptic equations with multiple solutions \ast . / Wang, Yingwei; Hao, Wenrui; Lin, Guang.

In: SIAM Journal on Scientific Computing, Vol. 40, No. 4, 01.01.2018, p. B1180-B1205.

Research output: Contribution to journalArticle

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