Advection-diffusion-reaction (ADR) problems

  • Luca Formaggia
  • Fausto Saleri
  • Alessandro Veneziani
Part of the UNITEXT book series (UNITEXT)


We have seen in  Chapter 3 that Galerkin method applied to elliptic problems in the form: find uVH 1 (Ω) such that
$$ a\left( {u,v} \right) = F\left( v \right) \forall v \in V, $$
provides a convergent solution in the H 1(Ω) norm that satisfies
$$ {\left\| {{u_h}} \right\|_V} \leqslant \frac{M}{\alpha }, {\left\| {u - {u_h}} \right\|_{{H^1}(\Omega )}} \leqslant \frac{\gamma }{\alpha }\mathop {\inf }\limits_{{v_h} \in {V_h}} {\left\| {u - {u_h}} \right\|_{{H^1}(\Omega ),}} $$
where M is the continuity constant of F(·), α and γ coercivity and continuity constants of a(·,·) respectively. In practice, these inequalities can be meaningless when the constants involved are large. In particular if γ ≫ α the second inequality is an effective bound for the error only if \(\mathop {\inf }\limits_{{v_h} \in {V_h}} {\left\| {u - {v_h}} \right\|_{{H^1}(\Omega )}}\) is small. For a finite element discretization, this corresponds to a small value of the mesh size h. The associated discretized problem can be therefore computationally expensive or even not affordable.


Mass Matrix Galerkin Method Const Real Finite Element Discretization Convergent Solution 
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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Luca Formaggia
    • 1
  • Fausto Saleri
    • 1
  • Alessandro Veneziani
    • 2
  1. 1.MOX — Dipartimento di Matematica “F. Brioschi”Politecnico di MilanoItaly
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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