Bolstad, John H. and Keller, Herbert B. (1986) A multigrid continuation method for elliptic problems with folds. SIAM Journal on Scientific and Statistical Computing, 7 (4). pp. 1081-1104. ISSN 0196-5204. doi:10.1137/0907074. https://resolver.caltech.edu/CaltechAUTHORS:20120627-134928372
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Abstract
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0. For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points.
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| Additional Information: | © 1986 Society for Industrial and Applied Mamthematics. Received by the editors February 8, 1984, and in revised form June 13, 1985. This work was typeset at the Lawrence Livermore National Laboratory using a troff program running under UNIX. The final copy was produced on July 23, 1985. It was supported by the U.S. Department of Energy under contract EY-76-S-03-070 and by the U.S. Army Research Office under contract DAAG-29-78-C-0011. We wish to thank Achi Brandt for technical discussions. He discovered this method before us but has not published it. We are also grateful for the use of computer time on the Fluid Dynamics VAX and the Applied Math-IBM 4341 at Caltech. The former is supported by the Office of Naval Research, and the latter is supported by the IBM Corporation. Finally, we thank the referees for carefully reading the manuscript. | ||||||||||
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| Subject Keywords: | multigrid; arc-length continuation, nonlinear elliptic eigenvalue problems; limit points; folds; frozen tau method; tau extrapolation; deferred correction; defect correction | ||||||||||
| Issue or Number: | 4 | ||||||||||
| Classification Code: | AMS(MOS) subject classifications: 65N20; 65N25 | ||||||||||
| DOI: | 10.1137/0907074 | ||||||||||
| Record Number: | CaltechAUTHORS:20120627-134928372 | ||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20120627-134928372 | ||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||
| ID Code: | 32150 | ||||||||||
| Collection: | CaltechAUTHORS | ||||||||||
| Deposited By: | INVALID USER | ||||||||||
| Deposited On: | 27 Jun 2012 21:45 | ||||||||||
| Last Modified: | 09 Nov 2021 21:25 |
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