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

$$ a\left( {u,v} \right) = F\left( v \right) \forall v \in V, $$

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.