330 APPENDIX C A Cheaper Way to Complementarity
Alas, this nice approach has a serious drawback: As we saw in
Section 6.3, the part of the computation that yields a is much more
expensive 1 than the determination of q0.
Therefore, it would be interesting to be able to save on this effort, in
the case of the a-method, by making good use of the information one
has, once h m has been determined. In quite fuzzy terms for the
moment--but this will become more and more precise--can the solution
in terms of q~ somewhat be corrected to yield a truly solenoidal (not
only m-solenoidal) approximation of b?
C.1 LOCAL CORRECTIONS
So let's suppose we have computed h e satisfying Eqs. (1), (2), and (6),
and such that
(8)
~D~ h. gradKn= 0 V n
~ Nh ,
where ;~n (preferred in this Appendix to w n, for notational uniformity)
is the barycentric function of node n, and Nh = N- N (S h) the set of
nodes not included in S h. We want some b ~ Wam, divergence-free,
andmin order to make use of the knowledge of the solution we have
already acquiredmclose to h.
What we have done in Chapter 6 seems to give an obvious solution:
Look for a minimizer of the error in constitutive law,
(9) b = arginf{~ D
~-1 I b'-
~h m
I 2"
b'
~ IBFm },
where IBFm
-- {b E
W2(D) " div b = 0, n. b = 0 on S b Jc n. b = F}.
Vector fields in this space are linear combinations of face elements,
(10) b -- ~f~rb bf wf,
where YB abbreviates F(S- sD), the set of faces not included in S b. (This
way, (10) implicitly takes the no-flux condition (4) into account.) But the
remaining nonzero face-DoFs bf are not independent for b
~ IB F .
They
m
are constrained by linear relations:
1just for practice, let's do it again, this time with the ratio T/N equal to 6. Thanks to
the Euler-Poincar6 formula, one has E ~ 7N and F ~ 12 N. The average number of faces
that contain a given edge is 3F/E, so each edge has 9F/E "neighbors", if one defines as
neighbors two edges that belong to a common tetrahedron. The number of off-diagonal
entries of the edge-element matrix is thus 9F, that is, 108 N, against 14 N for the matrix
created by the q~-method.
C.1 LOCAL CORRECTIONS 331
~2f~
F(T)
DTf
bf = 0, ~f~ y(c)Dcf bf = F,
where F(C) is the set of faces that pave the cut C, and Dcf= + 1
according to relative orientation. As IB F = rot A F (9) is equivalent to
m m !
finding a minimizer
(11)
am ~ arginf{~D ~1 I rot a-~h m [2. a ~ AF},
and there is no difference between doing that and directly solving for a
by edge elements. The b = rot a thus obtained, which is the approximation
b m of Chapter 6, is indeed the closest to h m in energy. But no use is
made of the knowledge of h this way.
Remark C.1. Problem (11) is the same as problem (6.21)" Since
YD
[.~-1 I rot a - ,h
I 2 -- YD
,-1 I rot
a I 2 -
2~ D
h m. rot a
+
YD " I hm
12
and (by
Lemma 6.1)
fD
hm" rot a = 0, the two functionals in (11) and (6.21) differ
by a constant, and minimization is performed on the same subspace. ~)
We now introduce the localization heuristics. Let's have a Partition of
unity over D, i.e., a family of piecewise smooth functions Z', indexed
over some finite set y, and satisfying ~ i~ ~ Z i = 1. Any b can be written
as a sum b = ~ i ~ ~ Z i b - ~
i
~ b 1. For one that suits our needs (divergence
free, and close to h), each b i should satisfy div bi= div(zlb) =
b . grad Z i, and should be close to Zi/Jhm . Not knowing b, we replace
b . grad Z i by the next best thing, which is ~th m . grad ~i, and try to
achieve div b I = ~h m . grad Z i as best we can, while looking for
b i
in W 2.
Since then div b' belongs to W 3, and is thus mesh-wise constant, the
best we can do is to request
(12)
~T div
bi--
fT ~1 h.
grad
X i ~/
T E T,
for all indices i. Besides that, we also want b i as close as possible, in
energy, to zi~th, hence
(13)
bi=
arginf{fD
[.1-1 I b'-
[J, Xihm [2. b' ~ IBi(hm)},
where IBi(hm) is an ad-hoc and provisional notation for the set of bis in
IBFm
that satisfy the constraints (12). Intuitively (and we'll soon confirm
this), computing b i is a local procedure. (Notice that div b = div(~ i b i) =
0, by summing (12) over i.) This is the principle.
For its formal application, now, let us call T i the set of tetrahedra
whose intersection with supp(x i) has a nonzero measure, D i their set
union, and F i the collection of all faces of such tetrahedra except those
contained in the boundary 3D ~ (but not in S h, cf. Fig. C.1). In (13),
332 APPENDIX C A Cheaper Way to Complementarity
lzZ i h = 0 outside EY, so we may search
b i
among the restricted set of
fields that vanish outside D i, which means (since by normal continuity
of b i, its normal component on 3D i must be null) those of the form
~f~
yi
b f wf. Let us therefore introduce the notation
W2m(D i) = {b ~ warn(D) " b = ~
f~ F i
bf wf},
and redefine IBi(hm) as
IBi(hm) =
{b
~ IBF,, ~ W2m (Di)"
~T div
b i= YT
~th. grad Z i V
T E T i}.
The b~s are given by
(14) b i = arginf{~Di ~t -1 I b' _ _,~_.. Ii~/h [i 2. b' ~ IB~(h )},
which differs from (13) only by the integration domain being D i instead
of D.
D
S h 0 ......... ~:~',:~ii':iiii~i: ::":::::: ........ ii'~',:, ~,: ~
~ 0 ............................................ i% 0
0 i h
sb ~C
FIGURE C.1. Two examples showing the relation between supp(z i) (shaded)
and T i. Faces of Y~ (appearing as edges in this 2D drawing) are those not
marked with a 0.
Before going further, let us point to an easily overlooked difficulty:
If D i does not encounter S h (we call Yh the subset of Y for which this
happens),
then
~D i div
b i= 0. So
unless
(15)
~D ~h .gradz i=0 Vi~Yh,
some of the affine sets IBi(hm) may well be empty! Fortunately, there are
easy ways to enforce condition (15). One is to use the barycentric functions
as partition of unity, and then Yh coincides with Nh, so (15) is equivalent
to (8), which is indeed satisfied if h was computed by the q0-method.
More generally, if all the zis are linear combinations of the ~ns, which
we shall assume from now on, (15) holds, and we are clear.
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