Multiconstrained variational problems of nonlinear eigenvalue type: new formulations and algorithms

Authors:
Alexander Eydeland, Joel Spruck and Bruce Turkington

Journal:
Math. Comp. **55** (1990), 509-535

MSC:
Primary 49R05; Secondary 35J85, 76W05

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035931-7

MathSciNet review:
1035931

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new variational approach is proposed for a class of semilinear elliptic eigenvalue problems involving many eigenvalue parameters. These problems arise, for instance, in the modelling of magnetohydrodynamic equilibria with one spatial symmetry. In this case, the physical variational principle imposes a continuously infinite family of constraints, which prescribes the mass and helicity within every flux tube. The equilibrium equations therefore contain unspecified profile functions that are determined along with the solution as multipliers for those constraints. A prototype problem for this general class is formulated, and a natural discretization of its constraint family is introduced. The resulting multiconstrained minimization problem is solved by an iterative algorithm, which is based on relaxation of the given nonlinear equality constraints to linearized inequalities at each iteration. By appealing to convexity properties, the monotonicity and global convergence of this algorithm is proved. The explicit construction of the iterative sequence is obtained by a dual variational characterization.

**[1]**V. I. Arnold,*Variational principles for three-dimensional steady state flows of an ideal fluid*, Appl. Math. Mech.**29**(1965), 1002-1008.**[2]**V. I. Arnol′d,*Mathematical methods of classical mechanics*, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR**0690288****[3]**G. Bateman,*MHD instabilities*, MIT Press, Cambridge, Mass., 1978.**[4]**T. Brooke Benjamin,*Impulse, flow force and variational principles*, IMA J. Appl. Math.**32**(1984), no. 1-3, 3–68. MR**740456**, https://doi.org/10.1093/imamat/32.1-3.3**[5]**F. Bauer, O. Betancourt, and P. Garabedian,*A computational method in plasma physics*, Springer-Verlag, Berlin, Heidelberg, and New York, 1978.**[6]**R. Courant and D. Hilbert,*Methods of mathematical physics*, Vol. 1, Interscience, New York, 1937.**[7]**Alexander Eydeland and Joel Spruck,*The inverse power method for semilinear elliptic equations*, Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 273–286. MR**956070**, https://doi.org/10.1007/978-1-4613-9605-5_16**[8]**A. Eydeland, J. Spruck, and B. Turkington,*A computational method for multiconstrained variational problems in magnetohydrodynamics*(in preparation).**[9]**Avner Friedman,*Variational principles and free-boundary problems*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. MR**679313****[10]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR**0473443****[11]**R. Glowinski,*Numerical methods for nonlinear variational problems*, Springer-Verlag, New York, 1984.**[12]**Harold Grad,*Magnetic confinement fusion energy research*, Mathematical aspects of production and distribution of energy (Proc. Sympos., San Antonio, Tex., 1976) Proc. Sympos. Appl. Math., XXI, Amer. Math. Soc., Providence, R.I., 1977, pp. 3–40. MR**483655****[13]**H. Grad, P. N. Hu, and D. C. Stevens,*Adiabatic evolution of plasma equilibria*, Proc. Nat. Acad. Sci. U.S.A.**72**(1975), 3789-3793.**[14]**A. D. Ioffe and V. M. Tihomirov,*Theory of extremal problems*, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. MR**528295****[15]**M. D. Kruskal and R. M. Kulsrud,*Equilibrium of a magnetically confined plasma in a toroid*, Phys. Fluids**1**(1958), 265–274. MR**0112549**, https://doi.org/10.1063/1.1705884**[16]**Peter Laurence and E. W. Stredulinsky,*A new approach to queer differential equations*, Comm. Pure Appl. Math.**38**(1985), no. 3, 333–355. MR**784478**, https://doi.org/10.1002/cpa.3160380306**[17]**T. Ch. Mouschovias,*Nonhomologous contraction and equilibria of self-gravitating magnetic interstellar clouds embedded in an intercloud medium*:*Star formation*I,*Formulation of the problem and method of solution*, Astrophys. J.**206**(1976), 753-767.**[18]**J. Mossino and R. Temam,*Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics*, Duke Math. J.**48**(1981), no. 3, 475–495. MR**630581****[19]**R. T. Rockafellar,*Convex analysis*, Princeton Univ. Press, Princeton, N.J., 1970.**[20]**R. Temam,*Monotone rearrangement of a function and the Grad-Mercier equation of plasma physics*, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 83–98. MR**533163****[21]**L. Woltjer,*Hydromagnetic equilibrium. III. Axisymmetric incompressible media*, Astrophys. J**130**(1959), 400–404. MR**0110412**, https://doi.org/10.1086/146731**[22]**L. Woltjar,*Hydromagnetic equilibrium. IV. Axisymmetric compressible media*, Astrophys. J**130**(1959), 404–413. MR**0110413**, https://doi.org/10.1086/146732

Retrieve articles in *Mathematics of Computation*
with MSC:
49R05,
35J85,
76W05

Retrieve articles in all journals with MSC: 49R05, 35J85, 76W05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035931-7

Keywords:
Semilinear elliptic eigenvalue problems,
constrained minimization,
iterative procedure,
magnetohydrodynamics

Article copyright:
© Copyright 1990
American Mathematical Society