6.3 WHY NOT STANDARD ELEMENTS?
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mesh
m and
continuous (in all its three scalar components), q0 is
piecewise quadratic
and
differentiable, two hardly compatible conditions.
The space p2 of piecewise quadratic functions on m is generated by the
products
WnWm,,
where n and m span the nodal set N. Therefore,
grad ~ - ~ m,n ~ N CG
n
grad(w Wn), and since the products WnW m are not
differentiable, the normal component of this field has a nonzero jump
across all faces. (This jump is affine with respect to coordinates over a
face.) Demanding that all these jumps be 0 is a condition that considerably
constrains the c~'s (in practice, only globally quadratic ~, as opposed to
mesh-wise quadratic, will comply). Consequently, the kernel of rot in
AF(Ip 1) will be of very low dimension, and as a rule reduced to 0,
because of additional constraints imposed by the boundary conditions.
So unless the mesh is very special, one may expect a regular
M p,
whereas
the matrix M is singular, since W 1 contains gradients (cf. Prop. 5.4,
asserting that WID grad W°).
Good news? We'll see. Let us now look at the weak points of the
nodal vectorial method.
6.3.2 Accuracy is downgraded
When working with the same mesh, accuracy is downgraded with nodal
1 F 1
elements, because rot(Afm(IP )) C rot(A m(W )), with as a rule a
strict
inclusion. Minimizing over a smaller space will thus yield a less accurate
upper bound in the case of nodal elements, for a given mesh m. (Let's
omit the subscript m for what follows, as far as possible. Recall that N,
E, F, T refer to the number of nodes, etc., in the mesh.)
The inclusion results from this:
Proposition
6.4.
For a given mesh m, any field
u ~ IP 1
is sum of some field
in W ~ and of the gradient of some piecewise quadratic function, i.e.,
IP 1 c W 1 q-
grad
p2.
Proof.
Given u e IP 1, set u e
= ~e X" U,
for all e e E, and let v =
~e ~ ~ Ue We be the field in W 1 which has these circulations as edge DoFs.
Then, both u and v being linear with respect to coordinates, rot(u- v)
is piecewise constant. But its fluxes through faces are 0, by construction
(again, see Fig. 5.5), so it vanishes. Hence u = v + V% where q0 is such
that Vq0 be piecewise linear, that is, q0 e p2.
As a corollary, rot IP 1 c rot W 1, hence the inclusion, as far as (but this
is what we assumed for fairness) AFm(IP~)
c a w.
As a rule, this is strict
inclusion, because the dimension of rot IP ~ cannot exceed that of IP 1,