3. Properties of the residue estimator

We note	\({\eta }_{\text{EX}}(K)\) the exact error \({\parallel u-{u}_{h}\parallel }_{{H}^{1}(K)}\) on the element \(K\) (unknown at first)
and	\({\eta }_{\text{EX}}(\Omega )\) the exact global error \({\parallel u-{u}_{h}\parallel }_{{H}^{1}(O)}\)

We then have the following properties ([bib1]):

• regardless of element \(K\), the elementary error \(\eta (K)\) is increased by the exact local error (multiplied by a constant independent of triangulation),

Be \(\forall K\eta (K)\le {C}_{1}\times {\eta }_{\text{EX}}(K)\)

• the exact global error is increased by the global estimated error \(\eta (\Omega )\) (multiplied by a constant independent of \(T\))

Be \({\eta }_{\text{EX}}(\Omega )\le {C}_{2}\times \eta (\Omega )\)

The constants \({C}_{1}\text{et}{C}_{2}\) depend in principle on the type of finite element and on the boundary conditions of the problem. Kelly and Gago [bib3] proposed in 2D a constant \({C}_{2}\) that only depends on the degree \(p\) of the interpolation polynomial used:

\(\begin{array}{cc}{C}_{2}={(\frac{1}{\text{24}{p}^{2}})}^{1/2}\text{soit}& {C}_{2}=\frac{1}{\mathrm{2p}\sqrt{6}}\text{pour les TRIA3 et QUAD4}(\text{degré 1})\\ & {C}_{2}=\frac{1}{\mathrm{4p}\sqrt{6}}\text{pour les TRIA6 et QUAD8}(\text{degré 2})\end{array}\)

For 3D, we do not have a constant evaluation. However, it can be said that the global estimated error overestimates the exact global error in all cases. This result is not necessarily true at the local level.