Study of the transitory problem
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We recall the expression of the continuous problem in which we are interested. Find :math:`({u}^{\text{*}},{p}^{\text{*}})` such as

:math:`\mathrm{\{}\begin{array}{c}\mathrm{-}{\mathrm{\nabla }}^{\text{*}}\mathrm{\cdot }{\sigma }^{\text{'*}}({u}^{\text{*}})+b{\mathrm{\nabla }}^{\text{*}}{p}^{\text{*}}\mathrm{=}{f}^{\text{*}}\text{dans}\mathrm{[}\mathrm{0,}{T}^{\text{*}}\mathrm{]}\mathrm{\times }{\Omega }^{\text{*}}\\ {\mathrm{\partial }}_{t}^{\text{*}}(\frac{E}{M}{p}^{\text{*}}+b{\mathrm{\nabla }}^{\text{*}}\mathrm{\cdot }{u}^{\text{*}})\mathrm{-}\frac{E}{M}{\Delta }^{\text{*}}{p}^{\text{*}}\mathrm{=}{g}^{\text{*}}\text{dans}\mathrm{[}\mathrm{0,}{T}^{\text{*}}\mathrm{]}\mathrm{\times }{\Omega }^{\text{*}}\end{array}`**eq 3.1-1**

In the thesis [:ref:`1 <1>`], 2 families of error indicators for this problem were proposed. Only one was returned in Code_Aster, allowing the pressure error to be evaluated effectively.

We define the error estimators in space:


* Estimator for the hydraulic equation: for all :math:`m\in \left[\mathrm{1,}N\right]`,

:math:`\begin{array}{c}{\tau }_{m}{E}_{p\mathrm{,0}}^{m}=\sum _{K\in {T}_{h}}{\tau }_{m}{E}_{p\mathrm{,0},K}^{m}\\ =\sum _{K\in {T}_{h}}({\tau }_{m}{h}_{K}^{2}\frac{{E}^{2}}{{P}^{2}{L}^{\text{dim}}\mathrm{\kappa M}}{\parallel \frac{1}{M}{\delta }_{t}{p}_{h}^{m}+b\nabla \cdot ({\delta }_{t}{u}_{h}^{m})\parallel }_{\mathrm{0,}K}^{2}+{\tau }_{m}{h}_{K}\frac{{E}^{2}}{{P}^{2}{L}^{\text{dim}}{\text{κρ}}^{2}M}\sum _{F\in {F}_{K}^{i}}{\parallel \left[{M}_{\text{lq},h}^{m}\cdot n\right]\parallel }_{\mathrm{0,}F}^{2}\\ +{\tau }_{m}{h}_{K}\frac{{E}^{2}}{{P}^{2}{L}^{\text{dim}}{\text{κρ}}^{2}M}\sum _{F\in {F}_{K}^{\partial }\cap {\Gamma }_{N}^{H}}{\parallel {M}_{\text{lq},\text{nor}}^{m}-{M}_{\text{lq},h}^{m}\cdot n\parallel }_{\mathrm{0,}F}^{2})\end{array}`


* Estimators for the mechanical equation: for all :math:`m\in \left[\mathrm{1,}N\right]`,

:math:`\begin{array}{c}{E}_{u}^{m}=\sum _{K\in {T}_{h}}{E}_{u,K}^{m}\\ =\sum _{K\in {T}_{h}}({h}_{K}^{2}\frac{1}{{P}^{2}{L}^{\text{dim}}}{\parallel {f}^{m}+\nabla \cdot {\sigma }^{\text{'}}({u}_{h}^{m})\parallel }_{\mathrm{0,}K}^{2}+{h}_{K}\frac{1}{{P}^{2}{L}^{\text{dim}}}\sum _{F\in {F}_{K}^{i}}{\parallel \left[{\sigma }^{\text{'}}({u}_{h}^{m})\cdot n\right]\parallel }_{\mathrm{0,}F}^{2}\\ +{h}_{K}\frac{1}{{P}^{2}{L}^{\text{dim}}}\sum _{F\in {F}_{K}^{\partial }\cap {\Gamma }_{N}^{M}}{\parallel {\sigma }_{\text{nor}}^{m}-({\sigma }^{\text{'}}({u}_{h}^{m})\cdot n-{\text{bp}}_{h}^{m}n)\parallel }_{\mathrm{0,}F}^{2})\end{array}`

:math:`\begin{array}{c}{E}_{u}^{m}({\delta }_{t})=\sum _{K\in {T}_{h}}{E}_{u,K}^{m}({\delta }_{t})\\ \sum _{K\in {T}_{h}}({h}_{K}^{2}\frac{1}{{P}^{2}{L}^{\text{dim}}}{\parallel {f}^{m}-{f}^{m-1}+\nabla \cdot {\sigma }^{\text{'}}({u}_{h}^{m}-{u}_{h}^{m-1})-b\nabla ({p}_{h}^{m}-{p}_{h}^{m-1})\parallel }_{\mathrm{0,}K}^{2}\\ +{h}_{K}\frac{1}{{P}^{2}{L}^{\text{dim}}}\sum _{F\in {F}_{K}^{i}}{\parallel \left[{\sigma }^{\text{'}}({u}_{h}^{m}-{u}_{h}^{m-1})\cdot n\right]\parallel }_{\mathrm{0,}F}^{2}\\ +{h}_{K}\frac{1}{{P}^{2}{L}^{\text{dim}}}\sum _{F\in {F}_{K}^{\partial }\cap {\Gamma }_{N}^{M}}{\parallel {\sigma }_{\text{nor}}^{m}-{\sigma }_{\text{nor}}^{m-1}-({\sigma }^{\text{'}}({u}_{h}^{m}-{u}_{h}^{m-1})\cdot n-b({p}_{h}^{m}-{p}_{h}^{m-1})n)\parallel }_{\mathrm{0,}F}^{2}\end{array}`

We define the estimator in time

:math:`{E}_{\text{tim}}^{m}={\tau }_{m}\frac{E}{{P}^{2}{L}^{\text{dim}}{\rho }^{2}\kappa }{\parallel {M}_{\text{lq},h}^{m}-{M}_{\text{lq},h}^{m-1}\parallel }_{\mathrm{0,}\Omega }^{2}`

The following properties are available:

**Theorem 1** (Reliability) For all :math:`n\in \left[\mathrm{1,}N\right]`,

:math:`{\int }_{0}^{{t}_{n}}{\parallel (p-{p}_{\mathrm{h\tau }})(s)\parallel }_{d}^{2}+{\int }_{0}^{{t}_{n}}{\parallel (p-{\pi }^{0}{p}_{\mathrm{h\tau }})(s)\parallel }_{d}^{2}<\sum _{m=1}^{N}{\tau }_{m}{E}_{p\mathrm{,0}}^{m}+\underset{0\le m\le N}{\text{sup}}{E}_{u}^{m}+{(\sum _{m=1}^{N}{({E}_{u}^{m}({\delta }_{t}))}^{1/2})}^{2}+\sum _{m=1}^{N}{E}_{\text{tim}}^{m}`

The operator :math:`{\pi }^{0}` designates the projection operator on the functions that are constant piecewise in time, i.e. :math:`{\pi }^{0}{p}_{\mathrm{h\tau }}` equal to :math:`{p}_{h}^{n}` over :math:`{I}_{n}` for all :math:`n\in \left\{\mathrm{1,}\cdots ,N\right\}`.

**Theorem 2** (Optimality of the time indicator)

We have the following estimate

:math:`{E}_{\text{tim}}=\sum _{m=1}^{N}{E}_{\text{tim}}^{m}<{\int }_{0}^{T}{\parallel (p-{p}_{\mathrm{h\tau }})(s)\parallel }_{d}^{2}\text{ds}+{\int }_{0}^{T}{\parallel (p-{\pi }^{0}{p}_{\mathrm{h\tau }})(s)\parallel }_{d}^{2}\text{ds}`

**Theorem 3** (Optimality of indicators in space) For all :math:`K\in {T}_{h}`, we have

:math:`\begin{array}{c}{E}_{u,K}^{m}<\sum _{{K}^{\text{'}}\in {\Delta }_{K}}\left[{h}_{K}^{2}{\parallel {f}^{m}-{f}_{h}^{m}\parallel }_{\mathrm{0,}{K}^{\text{'}}}+{\parallel {u}^{m}-{u}_{h}^{m}\parallel }_{a,{K}^{\text{'}}}^{2}+{\parallel {p}^{m}-{p}_{h}^{m}\parallel }_{\mathrm{0,}{T}^{\text{'}}}^{2}\right]\\ {E}_{u,K}^{m}({\delta }_{t})<\sum _{{K}^{\text{'}}\in {\Delta }_{K}}{\tau }_{m}^{2}\left[{h}_{K}^{2}{\parallel {\delta }_{t}{f}^{m}-{\delta }_{t}{f}_{h}^{m}\parallel }_{\mathrm{0,}{K}^{\text{'}}}+{\parallel {\delta }_{t}{u}^{m}-{\delta }_{t}{u}_{h}^{m}\parallel }_{a,{K}^{\text{'}}}^{2}+{\parallel {\delta }_{t}{p}^{m}-{\delta }_{t}{p}_{h}^{m}\parallel }_{\mathrm{0,}{T}^{\text{'}}}^{2}\right]\\ {\tau }_{m}{E}_{p\mathrm{,0},K}^{m}<\sum _{{K}^{\text{'}}\in {\Delta }_{K}}{h}_{K}^{2}{\int }_{{I}_{m}}[{\parallel (g-{\pi }^{0}{g}_{\mathrm{h\tau }})(s)\parallel }_{\mathrm{0,}{K}^{\text{'}}}^{2}\text{ds}+{\tau }_{m}^{2}{\parallel {\delta }_{t}{u}^{m}-{\delta }_{t}{u}_{h}^{m}\parallel }_{a,{K}^{\text{'}}}\\ +{\tau }_{m}^{2}{\parallel {\delta }_{t}{p}^{m}-{\delta }_{t}{p}_{h}^{m}\parallel }_{\mathrm{0,}{K}^{\text{'}}}^{2}+{h}_{T}^{-2}{\parallel (p-{\pi }^{0}{p}_{\mathrm{h\tau }})(s)\parallel }_{d,{K}^{\text{'}}}^{2}]\text{ds}\end{array}`