ABSTRACT
We consider here, as a model problem, the classical toy problem of advection-
dominated linear equations. From the physical point of view, we may think
to the problem of the passive transport of a scalar diffusive quantity in a fluid
whose velocity is known. Let then ft be, for instance, a convex polygon, e a
positive number (= diffusion coefficient), c a bounded mapping from ft to IR2
(= velocity field) and /, say, an element of L2(ft) (= source term). We consider
then the problem of finding u in Hq (ft) such that
—sAu -f c • Vw = / in ft. (3-1)
We can set Lu — sAu + c • Vu, and
C(u, v) := ea(u, v) + c(ti, v) Vw, v E #¿(0), (3.2)
where, in a natural way,
a(u,v) := I V u V v d x , and c(u,v) := / c-Vuvdx Vu, v £ H q (ft). J J ci
Assume now that we are given a decomposition Th of ft into triangles, and
assume moreover that c and / are piecewise constant on Th-We take then Vh
to be the space of piecewise linear continuous functions vanishing on 5ft, and
Bh as in (2.4) with B(K) = Hq(K) for each K. If we apply the theory of the
previous section, the bubble equation (2.11) becomes, in each triangle K: find
ug k in Hq(K) such that
—sAub^k + c • Vub}k = —{—sAuh + c • V% ) + / in K. (3.4)
As we already pointed out, equation (3.4) is unsolvable. As we shall see, there
are ways to get around this difficulty in a more or less satisfactory way. Before
discussing that, however, we want to point out how the solution in the model
case can be used. In particular, it is not difficult to check that, in the present
case, we have a(uB,Vh) — 0 for every ub G Bh and for every Vh G V/>. Hence
the additional term (2.16) arising in (2.15) becomes
£{uB, vh) = c(uB, vh) = / c-'VuB vh dx = - uB c-Vvhdx, (3.5) Ja Jn
with an obvious integration by parts. We also remark that the term c • Vv/, is
piecewise constant. Hence we see that only the mean value of ub in each K
will be used in the final system (2.15) for computing Uh. Moreover, still in our
assumptions, we observe that the right-hand side of (3.4) is also constant in K,
so that ub,k, in each A", can be written as
(3.6)
where
Rk := -{-eAuh + c • Vuh) + / (3.7)
is the residual in K (taking Uh as approximate solution) and the bubble bx is
the solution of the scaled problem:
find bx G H q(K) such that :
(3.8)
A simple computation shows that, inserting (3.6) in (3.5), the additional term
(2.16) becomes
c(uB,vh) = V ' [ (c • Vuft - /) C • Vvft dz, K I I
(3.9)
where bx is still the solution of (3.8), which is still unsolvable. This, as already
pointed out in [11] (see also [24], [5],) corresponds to the use of the well known
SUPG (Streamline Upwind Petrov Galerkin) method (see [12], [14]) with the
stabilising parameter chosen as
We still have to tackle the problem of getting an approximate solution of (3.8).