Advection-diffusion-reaction (ADR) problems

Abstract

We have seen in Chapter 3 that Galerkin method applied to elliptic problems in the form: find u ∈ V ⊆ H 1 (Ω) such that (Formula presented.) provides a convergent solution in the H 1(Ω) norm that satisfies (Formula presented.) 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 (Formula presented.) 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.