Abstract
In this paper, motivated by some recent progress in the development of Partial Newton-Correction Method for finding multiple solutions to nonlinear PDEs and by a closer observation to the implementation of the Local Min-Orthogonal (LMO) method developed in 2004, first a new L-\(\bot \) selection expression is introduced, then the (strict) separation condition and the continuity condition used in the mathematical framework of the LMO method are successively improved or weakened so that they are not only closer to the real algorithm’s implementation but also able to improve the relevant analysis. A new step-size rule and a new local characterization on saddle points are then established, based on which an improved LMO method is developed. The results in the paper can be further used to improve analysis of relevant numerical methods for solving multiple-solution problems in nonlinear PDEs. Finally, two numerical examples are carried out to illustrate the effectiveness of the new method and some new numerical findings are also presented.
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The datasets generated during the current study are available from the corresponding author on reasonable request.
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Funding
The second author was supported in part by NNSF of China (Nos.11871043,12171322), the Science and Technology Innovation Plan of Shanghai, China (No.20JC1414200) and the NSF of Shanghai, China (Nos. 21ZR1447200,22ZR1445500).
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Chen, X., Li, Z. & Zhou, J. An Improved Local-Min-Orthogonal Method for Finding Multiple Solutions to Nonlinear Elliptic PDEs. J Sci Comput 92, 1 (2022). https://doi.org/10.1007/s10915-022-01842-7
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DOI: https://doi.org/10.1007/s10915-022-01842-7
Keywords
- Local-min-orthogonal method
- (Directional) L-\(\bot \)
- Saddle points
- Multiple unstable solutions
- Henon’s equation
- Nonlinear PDEs