REFERENCES 189
also a surface operator by the same argument. The formula
~s Us" grads q°s = -~s q°s divs Us + ~s q°s v. u s
(where v now refers to the rim of S), is proved by the same technique as
in dimension 3, but (35) suggests to also introduce a rot operator acting
on
scalar
surface fields, as follows:
rot s q0 s = - n x grad s q0s,
hence the formula
~s q°s rots Us = ~s Us" r°tsq°s- ~s q°s I:. u S,
the nice symmetry of which compensates for the slight inconvenience s
of overloading the symbol rot s .
REFERENCES and Bibliographical Comments
The search for complementarity, a standing concern in computational electro-
magnetism [HP, HT], is related to the old "hypercircle" idea of Prager and Synge
[Sy]. In very general terms, this method consisted in partitioning the set of
equations and boundary conditions into two parts, thus defining two orthogonal
subsets in the solution space, the solution being at their intersection. Finding
two approximations within each of these subsets allowed one (by the equivalent
of the Pythagoras theorem in the infinite dimensional solution space, as was
done above in 6.3.1) to find the center and radius of a "hypercircle" containing
the unknown solution, and hence the bounds (not only on quadratic quantities
such as the reluctance, but on linear functionals and, even better, on pointwise
quantities [Gr]). After a period of keen interest, the idea was partly forgotten,
then revisited or rediscovered by several authors [DM, HP, LL, Ny, R&, Va .... ],
Noble in particular [No], who is credited for it by some (cf. Rall [Ra], or Arthurs
[An], who also devoted a book to the subject JAr]). Thanks to the Whitney
elements technology, we may nowadays adopt a different (more symmetrical)
partitioning of equations than the one performed by Synge on the Laplace problem.
(The one exposed here was first proposed in [B1].) On pointwise estimates,
which seem to stem from Friedrichs [Fr], see [Ba], [Co], [Ma], [St], [Sn].
8The risk of confusion, not to be lightly dismissed, will be alleviated by careful definition
of the types of the fields involved: The first rot is
VECTOR ~ SCALAR
(fields), the other
one is
SCALAR --> VECTOR.
Various devices have been proposed to make the distinction,
including the opposition rot vs Rot, but they don't seem to make things more mnemonic.
A. Di Carlo has proposed to denote the second operator by "grot", which would solve this
terminological difficulty.
190
CHAPTER 6 The "Curl Side": Complementarity
[A&]
[AE]
[An]
[Ar]
[Br]
[BC]
[Ba]
[B1]
[B2]
[B3]
[B41
[Bw]
[Col
[DM]
[DB]
[D&]
[Fe]
[Fr]
[Fr]
R. Albanese, R. Fresa, R. Martone, G. Rubinacci: "An Error Based Approach to the
Solution of Full Maxwell Equations", IEEE Trans., MAG-30, 5 (1994), pp.
2969-2971.
P. Alfeld, D.J. Eyre: "The Exact Analysis of Sparse Rectangular Linear Systems",
ACM TOMS, 17, 4 (1991), pp. 502-518.
N. Anderson, A.M. Arthurs, P.D. Robinson: "Pairs of Complementary Variational
Principles", J. Inst. Math. & Applications, 5 (1969), pp. 422-431.
A.M. Arthurs:
Complementary Variational Principles,
Clarendon Press (Oxford),
1970.
M.L. Barton: Tangentially Continuous Vector Finite Elements for Non-linear
3-D
Magnetic Field Problems,
Ph.D. Thesis, Carnegie-Mellon University
(Pittsburgh), 1987.
M.L. Barton, Z.J. Cendes: "New vector finite elements for three dimensional magnetic
fields computations", J. Appl. Phys., 61, 8 (1987), pp. 3919-3921.
N.M. Basu: "On an application of the new methods of the calculus of variations to
some problems in the theory of elasticity", Phil. Mag., 7,10 (1930), pp. 886-896.
A. Bossavit: "Bilateral Bounds for Reluctance in Magnetostatics", in
Numerical
Field Calculation in Electrical Engineering
(Proc. 3d Int. IGTE Symp.), IGTE (26
Kopernikusgasse, Graz, Austria), 1988, pp. 151-156.
A. Bossavit: "A Numerical Approach to Transient 3D Non-linear Eddy-current
Problems", Int. J.
Applied Electromagnetics in Materials,
1, 1 (1990), pp. 65-75.
A. Bossavit: "Complementarity in Non-Linear Magnetostatics: Bilateral Bounds
on the Flux-Current Characteristic", COMPEL, 11, 1 (1992), pp. 9-12.
A. Bossavit: "A new rationale for edge-elements", Int.
Compumag Society
Newsletter,
1, 3 (1995), pp. 3-6.
K. Bowden: "On general physical systems theories", Int. J.
General Systems,
18
(1990), pp. 61-79.
Ph. Cooperman: "An extension of the method of Trefftz for finding local bounds
on the solutions of boundary value problems, and on their derivatives",
Quart.
Appl. Math., 10, 4 (1953), pp. 359-373.
Ph. Destuynder, B. M6tivet: "Estimation explicite de l'erreur pour une m6thode
d'616ments finis non conforme", C.R. Acad. Sci. Paris, S4rie I (1996), pp. 1081-1086.
D.C. Dibben, R. Metaxas: "A Comparison of the Errors Obtained with Whitney
and Linear Edge Elements", IEEE Trans., MAG-33, 2 (1997), pp. 1524-1527.
P. Dular, J.-Y. Hody, A. Nicolet, A. Genon, W. Legros: "Mixed Finite Elements
Associated with a Collection of Tetrahedra, Hexahedra and Prisms", IEEE
Trans.,
MAG-30, 5 (1994), pp. 2980-2983.
W. Fenchel: "On Conjugate Convex Functions", Canadian J. Math., 1 (1949), pp.
23-27.
R.L. Ferrari: "Complementary variational formulations for eddy-current problems
using the field variables E and H directly", IEE Proc., Pt. A, 132, 4 (1985), pp.
157-164.
K.O. Friedrichs: "Ein Verfahren der Variationsrechnung, das Minimum eines
Integrals als das Maximum eines Anderen Ausdruckes darzustellen',
Nachrichten
der
Ges. d.
Wiss. zu Gi~ttingen
(1929), p. 212.
REFERENCES
191
[Gr]
[HP]
[HT]
[Ja]
[Ke]
[Ko]
[LL]
[Ma]
[Mk]
[MS]
[Mo]
[Ny]
[Ne]
[No]
[OR]
[P&]
[PFI
[PT]
[Ra]
[Re]
H.J. Greenberg: "The determination of upper and lower bounds for the solution of
the Dirichlet problem", J. Math. Phys., 27 (1948), pp. 161-182.
P. Hammond, J. Penman: "Calculation of inductance and capacitance by means of
dual energy principles", Proc. IEE, 123, 6 (1976), pp. 554-559.
P. Hammond, T.D. Tsiboukis: "Dual finite-elements calculations for static electric
and magnetic fields", IEE Pro¢., 130, Pt. A, 3 (1983), pp. 105-111.
D.A.H. Jacobs: "Generalization of the Conjugate Gradient Method for Solving
Non-symmetric and Complex Systems of Algebraic Equations", CEGB Report
RD/L/N70/80 (1980).
L. Kettunen, K. Forsman, D. Levine, W. Gropp: "Volume integral equations in
nonlinear 3-D magnetostatics', Int. J. Numer. Meth. Engng., 38, 16 (1995), pp.
2655-2675.
P.R. Kotiuga: "Analysis of finite-element matrices arising from discretizations of
helicity functionals", J. Appl. Phys., 67, 9 (1990), pp. 5815-5817.
P. Lad6v6ze, D. Leguillon: "Error estimate procedure in the finite element method
and applications", SIAM J. Numer. Anal., 20, 3 (1983), pp. 485-509.
C.G. Maple: "The Dirichlet problem: Bounds at a point for the solution and its
derivative", Quart. Appl. Math., 8, 3 (1950), pp. 213-228.
P. Monk: "A finite element method for approximating the time-harmonic Maxwell
equations", Numer. Math., 63 (1992), pp. 243-261.
P. Monk, E. Sfili: "A convergence analysis of Yee's scheme on nonuniform grids",
SIAM J. Numer. Anal., 31, 2 (1994), pp. 393-412.
J.J. Moreau: Fonctionnelles convexes, S6minaire Leray, Coll6ge de France, Paris
(1966).
B. Nayroles: "Quelques applications variationnelles de la th6orie des fonctions
duales h la m6canique des solides", J. de M6canique, 10, 2 (1971), pp. 263-289.
J.C. Nedelec: "A new family of mixed finite elements in IR 3 " Numer. Math., 50
(1986), pp. 57-81.
B. Noble: "Complementary variational principles for boundary value problems I:
Basic principles with an application to ordinary differential equations", MRC
Technical Summary Report N ° 473, Nov. 1964.
J.T. Oden, J.N. Reddy: "On Dual Complementary Variational Principles in
Mathematical Physics", Int. J. Engng. Sci., 12, 1 (1974), pp. 1-29.
J. Parker, R.D. Ferraro, P.C. Liewer: "Comparing 3D Finite Element Formulations
Modeling Scattering from a Conducting Sphere", IEEE Trans., MAG-29, 2 (1993),
pp. 1646-1649.
J. Penman, J.R. Fraser: "Dual and Complementary Energy Methods in Electro-
magnetism", IEEE Trans., MAG-19, 6 (1983), pp. 2311-2316.
Y. Perr6al, P. Trouv6: "Les m6thodes variationnelles pour l'analyse et l'approxi-
mation des 6quations de la physique math6matique: Les m6thodes mixtes-
hybrides conservatives", Revue Technique Thomson-CSF, 23, 2 (1991), pp.
391-468.
L.B. Rall: "On Complementary Variational Principles", J. Math. Anal. & Appl., 14
(1966), pp. 174-184.
Z. Ren, in A. Bossavit, P. Chaussecourte (eds.): The TEAM Workshop in Aix-les-
Bains, July 7-8 1994, EdF, Dpt MMN (1 Av. Gal de Gaulle, 92141 Clamart), 1994.
192
CHAPTER 6 The "Curl Side": Complementarity
[R§]
[Ro]
[Rt]
[St]
[Sn]
[Sy]
[To]
[Ts]
[Va]
J. Rikabi, C.F. Bryant, E.M. Freeman: "An Error-Based Approach to Complementary
Formulations of Static Field Solutions", Int. J.
Numer. Meth.
Engnrg., 26 (1988),
pp. 1963-1987.
R.T. Rockafellar: Convex Analysis, Princeton U.P. (Princeton), 1970.
J.P. Roth: "An application of algebraic topology: Kron's method of tearing",
Quart. Appl.
Math., 17, 1 (1959), pp. 1-24.
H. Stumpf: "Uber punktweise Eingrenzung in der Elastizit~itstheorie. I",
Bull.
Acad. Polonaise Sc., S6r. Sc. Techniques, 16, 7 (1968), pp. 329-344, 569-584.
J.L. Synge: "Pointwise bounds for the solutions of certain boundary-value problems",
Proc. Roy. Soc. London, A 208 (1951), pp. 170-175.
J.L. Synge:
The Hypercircle in Mathematical
Physics, Cambridge University Press
(Cambridge, UK), 1957.
E. Tonti: "On the mathematical structure of a large class of physical theories",
Lincei, Rend.
Sc. fis. mat. e nat., 52,1 (1972), pp. 51-56.
I.A. Tsukerman: "Node and Edge Element Approximation of Discontinuous Fields
and Potentials", IEEE Trans., MAG-29, 6 (1993), pp. 2368-2370.
M.N. Vainberg: Variational Methods for the Study of Nonlinear Operator
Equations,
Holden Day (San Francisco), 1963. (Russian edition: Moscow, 1956.)
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