REFERENCES
217
delicate point is the computation of the divergence
in the sense of distributions
of the field x -4 - x/Ixl
3
(refer to A.1.9 for the arrowed notation).
SOLUTIONS
7.1. Co~(D), which is dense in L2(D) with respect to the L2-norm, is
indeed dense also in the subspace
L2grad(W),
with
respect to this same norm.
But with respect to the
stronger
norm put
on
Lagrad(D), which is the point,
it's not, simply because there are
fewer
Cauchy sequences for this norm,
so their limits form a smaller space, namely, H10(D). In short, the stronger
the norm, the smaller the closure.
7.2. Let
q)n =
Y -4 n (1
- n 4
l y
- x n [+),
where + denotes the positive part
-2
of an expression. Then j lgrad
q)n ] 2
is in n , so q0
= ~n q)n is
in the
Beppo Levi space, but q)(x n) = n for n large enough, and doesn't vanish
at infinity.
7.3. See Ref. [CC].
7.5. The same computation as in Exer. 4.9 would give
(*) div(x
-4 X/ [X[3) -- X -4
3/1X13-3
X .X/I
x 15-0,
if it were not for the singularity at the origin. What was obtained there is
only the "function part" of the
distribution
div(x -4 x/Ix13), which is
therefore concentrated at the origin. To find it, apply Ostrogradskii to a
sphere of radius r centered at the origin, which gives the correct result,
div(x --9 x/l x] 3) = - 4r~ 80. Then, denoting by Z the kernel x --9
1 / (4r~ I xl ), one has q0 = Z * q, and hence, - Aqo = - A(K * q) = (- AZ) * q =
80" q=q.
REFERENCES
[Ad]
[AL]
[AR]
[B&]
[B21
R. Adams: Sobolev Spaces, Academic Press (Orlando, FL), 1975.
B.I. Agoshkov, B.I. Lebedev: "Operatory Puancare-Steklova i metody razdeleniya
oblasti v variatsionnyx zadachax", in Vychislitel'nye Protsessy i Sistemy, Nauka
(Moscow), 1985, pp. 173-227.
R. Albanese, G. Rubinacci: "Integral formulation for 3d eddy-current computation
using edge elements", IEE Proc., 135, Pt. A, 7 (1988), pp. 457-463.
A. Bossavit, C. Emson, I. Mayergoyz: M6thodes
num4riques en
41ectromagn6tisme,
Eyrolles (Paris), 1991.
A. Bossavit: "Mixed Methods and the Marriage Between 'Mixed' Finite Elements
and Boundary Elements", Numer. Meth. for PDEs, 7 (1991), pp. 347-362.
218
CHAPTER 7 Infinite Domains
[B3]
[Br]
[BM]
[cc]
[C1]
[Co]
[DL]
[Hu]
[IM]
[KP]
[LM]
[LS]
[Ma]
[MW]
[R&]
[RR]
[Tz]
[Yo]
[ZK]
A. Bossavit: "On various representations of fields by potentials and their use in
boundary integral methods", COMPEL, 9 (1990), Supplement A, pp. 31-36.
H. Brezis:
Analyse fonctionnelle,
Masson (Paris), 1983.
X. Brunotte, G. Meunier, J.F. Imhoff: "Finite element solution of unbounded problems
using transformations", IEEE Trans., MAG-30, 5 (1994), pp. 2964-2967.
P. Chaussecourte, C. Chavant: "Discretization of the Poincar6 Steklov operator
and condition number of the corresponding matrix in the Trifou FEM/BIEM
aproach to solving 3D eddy current equations", in A. Kost, K. Richter (eds.):
Confererence record, Compumag 1995
(TU Berlin, 1995), pp. 678-679.
C.J. Collie: "Magnetic fields and potentials of linearly varying current or
magnetisation in a plane bounded region", in
Proc. Compumag Conference on
the Computation of Magnetic fields,
St. Catherine's College, Oxford, March 31
to April 2, 1976, The Rutherford Laboratory (Chilton, Didcot, Oxon, U.K.), 1976,
pp. 86-95.
J.L. Coulomb: "A methodology for the determination of global electromechanical
quantities from a finite element analysis and its application to the evaluation of
magnetic forces, torques and stiffness", IEEE Trans., MAG-19, 6 (1983), pp. 2514-
2519.
R. Dautray, J.L. Lions (eds.):
Analyse math6matique et calcul num4rique pour
les sciences et les techniques,
t. 2, Masson (Paris), 1985.
M. Hulin, N. Hulin, D. Perrin"
]~quations de Maxwell, Ondes ]~lectromagn6tiques,
Dunod (Paris), 1992.
J.F. Imhoff, G. Meunier, J.C. Sabonnadi6re: "Finite element modeling of open
boundary problems", IEEE Trans., MAG-26, 2 (1990), pp. 588-591.
A. Kanarachos, C. Provatidis: "On the symmetrization of the BEM formulation",
Comp. Meth. Appl. Mech. Engng.,
71, 2 (1988), pp. 151-165.
J.L. Lions, E. Magenes"
Probl6mes aux limites non homog6nes et applications,
Vols. 1-2, Dunod (Paris), 1968.
H. Li, S. Saigal: "Mapped infinite elements for 3-D vector potential magnetic
problems", Int. J.
Numer. Meth. Engng., 37,
2 (1994), pp. 343-356.
I. Mayergoyz: "Boundary Galerkin's approach to the calculation of eddy currents
in homogeneous conductors", J. Appl. Phys., 55, 6 (1984), pp. 2192-2194.
B.H. McDonald, A. Wexler: "Finite element solution of unbounded field problems",
IEEE Trans., MTT-20, 12 (1972), pp. 841-347.
Z. Ren, F. Bouillault, A. Razek, J.C. V6rit6: "An efficient semi-analytical integration
procedure in three dimensional boundary integral method", COMPEL, 7, 4 (1988),
pp. 195-205.
Z. Ren, A. Razek: "Boundary Edge Elements and Spanning Tree Technique in
Three-Dimensional Electromagnetic Field Computation", Int. J.
Numer. Meth.
Engng.,
36, 17 (1993), pp. 2877-2693.
E. Trefftz: "Ein gegenstiick zum Ritzschen Verfahren", in
Proc. 2nd Int. Congr.
Appl. Mech. (Z/irich), 1926, pp. 131-137.
K. Yosida:
Functional Analysis,
Springer-Verlag (Berlin), 1965.
O.C. Zienkiewicz, D.W. Kelly, P. Bettess: "The Coupling of Finite Element and
Boundary Solution Procedures", Int. J.
Numer. Meth. Engng.,
11, 3 (1977), pp.
355-376.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.