Abstract
We examine Petviashvilli’s method for solving the equation \( \phi - \Delta \phi = |\phi |^{p-1} \phi \) on a bounded domain \(\Omega \subset \mathbb {R}^d\) with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on \(\mathbb {R}\) by Pelinovsky and Stepanyants in [16]. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.
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Notes
We use the definition \({{\mathrm{cn}}}= {{\mathrm{cn}}}(x;m)\) rather than \({{\mathrm{cn}}}= {{\mathrm{cn}}}(x; k^2)\).
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Acknowledgments
The authors are grateful for several helpful conversations with Svitlana Mayboroda. D.Olson was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. S.Shukla was supported by University of Minnesota UROP-11133. G.Simpson began this work under the support of the DOE DE-SC0002085 and the NSF PIRE OISE-0967140, and completed it under NSF DMS-1409018. D.Spirn was supported by NSF DMS-0955687.
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Olson, D., Shukla, S., Simpson, G. et al. Petviashvilli’s Method for the Dirichlet Problem. J Sci Comput 66, 296–320 (2016). https://doi.org/10.1007/s10915-015-0023-6
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DOI: https://doi.org/10.1007/s10915-015-0023-6