@article {25491,
	title = {Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems},
	journal = {SIAM Journal on Scientific Computing},
	volume = {39},
	number = {1},
	year = {2016},
	pages = {29-52},
	publisher = {SIAM},
	abstract = {This paper compares the performance of penalty and Lagrange multiplier approaches for the necessary unit-length constraint in the computation of liquid crystal equilibrium configurations. Building on previous work in [SIAM J. Sci. Comput., 37 (2015), pp. S157--S176;\&nbsp;SIAM J. Numer. Anal., 53 (2015), pp. 2226--2254], the penalty method is derived and well-posedness of the linearizations within the nonlinear iteration is discussed. In addition, the paper considers the effects of tailored trust-region methods in the context of finite-element discretizations and nested iteration for both formulations. Such methods are aimed at increasing the efficiency and robustness of each algorithm{\textquoteright}s nonlinear iterations. Three representative elastic equilibrium problems are considered to examine each method{\textquoteright}s performance. The first two configurations have analytical expressions for their exact solutions and, therefore, convergence to the true solution is considered. The third problem considers complicated boundary conditions, relevant in ongoing research, simulating surface nano-patterning. Finally, a novel multigrid scheme is introduced and tested for electrically and flexoelectrically coupled models to establish scalability for highly complicated applications. The Lagrange multiplier method is found to outperform the penalty method in a number of measures, the developed trust regions are shown to improve robustness, and nested iteration proves highly effective at reducing computational costs.\&nbsp;},
	doi = {10.1137/16M1058728},
	author = {James Haley {Adler} and David B. {Emerson} and MacLachlan, Scott and Thomas A. {Manteuffel}}
}
