The Mathematical Institute, University of Oxford, Eprints Archive

Deflation techniques for finding distinct solutions of nonlinear partial differential equations

Farrell, P. E. and Birkisson, A. and Funke, S. W. (2014) Deflation techniques for finding distinct solutions of nonlinear partial differential equations. Technical Report. Not specified. (Submitted)

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Abstract

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton-Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations are observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and
fluid mechanics that permit distinct solutions.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1858
Deposited By: Helen Taylor
Deposited On:17 Sep 2014 06:53
Last Modified:29 May 2015 19:33

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