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Erschienen in:
2010 | OriginalPaper | Buchkapitel
Finite Element and Finite Volume Methods
verfasst von : Torsten Linß
Erschienen in: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems
Verlag: Springer Berlin Heidelberg
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In this chapter we consider finite element and finite volume discretisations of
$$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1), \,\,\ u(0) = u (1) = 0,$$
with
b
≥ β > 0. Its associated variational formulation is: Find
$$u \in H_0^1 (0,1)$$
such that
(5.2)
$$a(u, v) = f(v)\,\,\, {\rm for\, all}\,\, v \in H_0^1 (0,1),$$
where
$$a(u,v): = \;\varepsilon (u',v') - (bu',v) + (cu,v)$$
and
(5.3)
$$f(v):=(f,v):= \int_0^1 {(fv)(x)dx.}$$
Throughout assume that
(5.4)
$$c + b' / 2 \ge \gamma > 0.$$
This condition guaranties the coercivity of the bilinear form in (5.2):
$$\||v |\|_\varepsilon^2 := _\varepsilon \|v' \|_0^2 + \gamma \|v \|^2_0 \le a(u,v)\,\,\,\, {\rm for\, all}\,\,\,\, v \in H^1_0 (0, 1).$$
This is verified using standard arguments, see e.g. [141]. If
b
≥ β > 0 then (5.4) can always be ensured by a transformation
$$\bar u(x) = u(x)e^{\delta x}$$
with δ chosen appropriately. We assume this transformation has been carried out.