A model problem

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).