Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T13:24:41.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Approximations of the Relative Rearrangement:The piecewise linear case.Application to some Nonlocal Problems

Published online by Cambridge University Press:  15 April 2002

Jean-Michel Rakotoson  and
Maria Luisa Seoane
Jean-Michel Rakotoson
Affiliation:
Département de Mathématiques, Université de Poitiers, boulevard 3, Téléport 2, B.P. 179, 86960 Futuroscope, France.
Maria Luisa Seoane
Affiliation:
Departamento de Matemática Aplicada, Facultad de Matemáticas, Campus Universitario, 15706 Santiago de Compostela, Spain.
Get access

Abstract

We first prove an abstract result for a class of nonlocalproblems using fixed point method. We apply this result toequations revelant from plasma physic problems. These equationscontain terms like monotone or relative rearrangement of functions.So, we start the approximation study by using finite element todiscretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

Keywords

Monotone decreasing and relative rearrangementsnonlocal problemsnumerical approximations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 2: Special issue for R. Teman's 60th birthday , March 2000 , pp. 477 - 499
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almgren, F. and Lieb, E., Symmetric rearrangement is sometimes continuous. J. Amer. Math. Soc. 2 (1989) 683-772. CrossRef
Beretta, E. and Vogelius, M., Symmetric rearrangement is sometimes continuous, An inverse problem originating from Magnetohydrodynamics II: the case of the Grad-Shafranov equation. Indiana University Mathematics Journal 41 (1992) 1081-1117. CrossRef
Berestycki, H. and Brezis, H., On a free boundary problem arising in plasma physics. Nonlinear Anal. 4 (1980) 415-436. CrossRef
Bermúdez, A. and Moreno, C., Duality methods for solving variational inequalities. Comp. and Math. Appl. 7 (1981) 43-58. CrossRef
Bermúdez, A. and Seoane, M.L., Numerical Solution of a Nonlocal Problem Arising in Plasma Physics. Mathematical and Computing Modelling. 27 (1998) 45-59. CrossRef
J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, Wiley, Gauthier-Villars (1989).
Blum, J., Gallouët, T. and Simon, J., Existence and Control of plasma equilibrium in a tokamak. SIAM J. Math. Anal. 17 (1986) 1158-1177. CrossRef
Boozer, A.H., Establishment of magnetic coordinates for given magnetic field. Phys. Fluids 25 (1982) 520-521. CrossRef
H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland (1973).
G. Chiti, Rearrangements of functions and convergence in Orlicz spaces. Applicable Analysis 9 (1979).
K.M. Chong and N.M. Rice, Equimesurable rearrangements of functions, Queen's University (1971).
P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambrigde University Press (1989).
Coron, J.M., The Continuity of the Rearrangement in $W^{1,p}({\mathbb R})$ . Annali della Scuola Normale Superiore di Pisa. Série IV 11 (1984) 57-85.
R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I., Interscience Pub. (1953).
J.I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators. Formulación global de las ecuacion es diferenciales no lineales y de las condiciones de contorno, CIEMAT, Informe #1 (1991).
J.I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators. Resultados de existencia de soluciones, CIEMAT, Informe #2 (1992).
J.I. Díaz, Modelos bidimensionales de equilibrio magnetohidrodinámico para Stellarators. Multiplicidad y dependencia de parámetros, CIEMAT, Informe #3 (1993).
Díaz, J.I. and J.M.Rakotoson, On a two-dimensional stationary free boundary problem arising in the confinement of a plasma in a Stellarator. C.R. Acad. Sci. Paris Serie I 317 (1993) 353-358.
Díaz, J.I. and Rakotoson, J.M., On a nonlocal stationary free boundary problem arising in the confinement of a plasma in a Stellarator geometry. Arch. Rat. Mech. Anal. 134 (1996) 53-95. CrossRef
I.Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland (1976).
Fernández-Cara, E. and Moreno, C., Critical Point Approximation Through Exact Regularization. Math. Comp. 50 (1988) 139-153. CrossRef
J.P. Freidberg, Ideal Magnetohydrodynamics, Plenum Press (1987).
A. Friedman, Variational principles and free-boundary problems, John Wiley and Sons (1982).
R. Glowinski, Numerical methods for non linear variational problems, Springer Verlag (1984).
H. Grad, Mathematical problem arising in plasmas physics. Proc. Intern. Congr. Math. Nice (1970).
J.M. Greene and J.L. Johnson, Determination of Hydromagnetic Equilibria. Phys. Fluids 27 (1984) 2101-2120
G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Cambridge University Press (1964).
Hender, T.C. and Carreras, B.A., Equilibrium calculation for helical axis Stellarators. Phys. Fluids 27 (1984) 2101-2120. CrossRef
B.Heron and M.Sermange, Non convex methods for computing free boundary equilibria of axially symmetric plasmas, Rapport de Recherche, I.N.R.I.A. (1981).
M.D. Kruskal and R.M. Kulsrud, Equilibrium of Magnetically Confined Plasma in a Toriod. Physics of Fluids 1, No. 4, (1958) 265-274.
A. Marrocco and O. Pironneau, Optimum desing with lagrangian finite elements: desing of an electromagnet, Rapport de Recherche, I.N.R.I.A. (1977).
Mignot, F. and Puel, J.P., On a class of nonlinear problems with positive, increasing, convex nonlinearity. Comm. Par. Diff. Eq. 5 (1980) 791-836. CrossRef
J. Mossin and J.M. Rakotoso, Isoperimetric inequalities in parabolic equations. Annali della Scuola Normale Superiore di Pisa. Série IV 13, No. 1, (1986) 51-73.
Mossino, J. and Temam, R., Directional Derivative of the Increasing Rearrangement Mapping and Application to a Queer Differential Equation in Plasma Physics. Duke Mathematical Journal 48 (1981) 475-495. CrossRef
J. Mossino and R. Temam, Free boundary problems in plasma physics, review of results and new developments. Free Boundary Problems: theory and applications. Vol I-II. Proc. Montec atini Symposium (1981). A. Fasano and M. Primicerio Eds, Pitman (1983) 672-681.
J. Mossino, Inégalités isopérmétriques et applications en physique, Hermann (1984).
K. Miyamoto, Plasma Physics for Nuclear Fusion, The M.I.T. Press (1987).
J.F. Padial, EDPs no lineales originadas en plasmas de fusión y filtración en medios porosos, Thesis Doctoral, Universidad Complutense de Madrid (1995).
J.F. Padial, J.M.Rakotoson and L. Tello, Introduction to the monotone and relative rearrangements and applications, Rapport, Département de Mathématiques, Université de Poitiers (1993).
G. Pòlya and W.N. Szegö, Isopermetric inequalities in mathematical physics, Princenton Univ. Press (1951).
Puel, J.P., A nonlinear eigenvalue problem with free boundary. C.R. Acad. Sci. Paris A 284 (1977) 861-863.
Rakotoson, J.M., Some properties of the relative rearrangement. J. Math. Anal. Appl. 135 (1988) 488-500. CrossRef
Rakotoson, J.M., A differentiability result for the relative rearrangement. Diff. Int. Eq. 2 (1989) 363-377.
J.M. Rakotoson, Relative rearrangement for highly nonlinear equations. Nonlinear Analysis. Theory, Meth. and Appl. 24 (1995) 493-507.
J.M. Rakotoson and M.L. Seoane (in preparation).
Rakotoson, J.M., Galerkin approximations, strong continuity of the relative rearrangement map and application to plasma physics equations. Diff. Int. Eq. 12 (1999) 67-81.
J.M. Rakotoson and B. Simon, Relative rearrangement on a measure space. Application to the regularity of weighted monotone rearrangement. Part I-II. Appl. Math. Lett. 6 (1993) 75-78; 79-92.
Rakotoson, J.M. and Simon, B., Relative rearrangement on a finite measure space. Application to weighted spaces and to P.D.E. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 91 (1997) 33-45.
Rakotoson, J.M. and Temam, R., A co-area formula with applications to monotone rearrangement and to regularity. Arch. Rational Mech. Anal. 109 (1991) 213-238. CrossRef
R.T. Rockafellar, Convex Analysis, Princeton University Press (1970).
V.D. Shafranov, On agneto-hydrodynamical equilibriium configurations. Soviet Physics JETP, 6 (1958) 5456-554.
G.G. Talenti, Rearrangements of functions and and Partial Differential Equations. Nonlinear Diffusion Problems, A. Fasano and M. Primicerio Eds, Springer-Verlag (1986) 153-178.
G.G. Talenti, Rearrangements and PDE. Inequalities, fifty years on from Hardy, Littlewood and Pòlya, W.N. Everitt Ed., Marcel Dekker Inc (1991) 211-230.
G.G. Talenti, Assembling a rearrangement. Arch. Rat. Mech. Anal. 98 (1987) 85-93
Temam, R., A nonlinear eigenvalue problem: equilibrium shape of a confined plasma. Arch. Rat. Mech. Anal. 65 (1975) 51-73. CrossRef
Temam, R., Remarks on a free boundary problem arising in plasma physics. Comm. Par. Diff. Eq. 2 (1977) 563-585. CrossRef
R.Temam, Monotone rearrangement of functions and the Grad-Mercier equation of plasma physics, Proc. Int. Conf. Recent Methods in Nonlinear Analysis and Applications, E. de Giogi and U. Mosco Eds (1978).
R.Temam, Analyse Numerique, Presses Universitaires de France (1971).
Toland, J.F., Duality in nonconvex optimization. J. Math. Appl. 66 (1978) 399-415.
Toland, J.F., Duality Principle, A for Non-convex Optimisation and the Calculus the Variations. Arch. Rat. Mech. Anal. 71 (1979) 41-61. CrossRef
R.S. Varga, Matrix Iterative Analysis, Prentice-Hall Inc. (1962)