Abstract
With the information provided by previously found solutions, an augmented singular transform is introduced in Xie et al. (JCAM 286:145–157, 2015) to change the local basin structure of the original problem for finding new solutions. However the old formulation in Xie et al. (JCAM 286:145–157, 2015) involves the kernel of an unknown solution to be found or the kernels of all previously found solutions, thus left several theoretical issues unsolved and prevents from further development. In this paper, we derive a new augmented singular transform which changes only the local basin/barrier structure around \(u=0\) for finding more solutions. Comparing to the old formulation, the new one is much easier to apply and resolves all unsolved theoretical issues left in Xie et al. (JCAM 286:145–157, 2015). A corresponding partial Newton-correction method is then designed to solve the augmented problem on the solution set. Mathematical justification of the new formulation, method and its local convergence are established. The new method is first tested on two very different variational problems and then applied to solve a nonvariational nonlinear convection-diffusion equation for multiple solutions, which are, for the first time, numerically computed and visualized with their profile and contour plots. Several interesting phenomena are observed for the first time and open for mathematical verification. Since the new formulation is general and simple, it can also be modified to treat other problems, e.g., quasilinear PDEs, a large system of PDEs with equality constraints, for finding multiple solutions.
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The authors wish to sincerely thank the anonymous reviewers for their very useful comments to improve the presentation of the paper.
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Z. Li: Supported in part by Innovation Program of Shanghai Municipal Education Commission, N.14YZ078 and NSF of Shanghai No.15ZR1430900.
Jianxin Zhou: Supported in part by NSF DMS-1115384.
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Li, Z., Wang, ZQ. & Zhou, J. A New Augmented Singular Transform and its Partial Newton-Correction Method for Finding More Solutions. J Sci Comput 71, 634–659 (2017). https://doi.org/10.1007/s10915-016-0314-6
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DOI: https://doi.org/10.1007/s10915-016-0314-6