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 language | English (US) |
|---|---|
| Pages (from-to) | B1180-B1205 |
| Journal | SIAM Journal on Scientific Computing |
| Volume | 40 |
| Issue number | 4 |
| DOIs | |
| State | Published - Jan 1 2018 |
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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 journal › Article
TY - JOUR
T1 - Two-level spectral methods for nonlinear elliptic equations with multiple solutions \ast
AU - Wang, Yingwei
AU - Hao, Wenrui
AU - Lin, Guang
PY - 2018/1/1
Y1 - 2018/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85053762556&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85053762556&partnerID=8YFLogxK
U2 - 10.1137/17M113767X
DO - 10.1137/17M113767X
M3 - Article
AN - SCOPUS:85053762556
VL - 40
SP - B1180-B1205
JO - SIAM Journal of Scientific Computing
JF - SIAM Journal of Scientific Computing
SN - 1064-8275
IS - 4
ER -