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Finite Element and Finite Volume Methods

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1985)

Abstract

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

$$a(u, v) = f(v)\,\,\, {\rm for\, all}\,\, v \in H_0^1 (0,1),$$
((5.2))

where

$$a(u,v): = \;\varepsilon (u',v') - (bu',v) + (cu,v)$$

and

$$f(v):=(f,v):= \int_0^1 {(fv)(x)dx.}$$
((5.3))

Throughout assume that

$$c + b' / 2 \ge \gamma > 0.$$
((5.4))

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.

Keywords

  • Bilinear Form
  • Finite Volume Method
  • Posteriori Error
  • Interpolation Error
  • Inverse Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Torsten Linß .

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© 2010 Springer-Verlag Berlin Heidelberg

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Linß, T. (2010). Finite Element and Finite Volume Methods. In: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics(), vol 1985. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05134-0_5

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