4. Pure residue indicator#

4.1. Ratings#

To build the local error indicator we will require the following notations:

  • The set of faces (resp. nodes) of the \(K\) element is designated by \(S(K)\) (resp. \(N(K)\)).

  • The set of nodes associated with one of its faces \(F\) (belonging to \(S(K)\)) is noted \(N(F)\).

Note:

To put it simply, the term « face » will refer to the side of a finite element in 2D or one of its faces in 3D.

  • The diameter of element \(K\) (or of one of its \(F\) faces) is denoted by \({h}_{K}\) (resp. \({h}_{F}\)).

  • The whole triangulation (\({T}_{h}\)) breaks down as

\({T}_{h}:={T}_{h,O}\cup {T}_{h\mathrm{,1}}\cup {T}_{h\mathrm{,2}}\cup {T}_{h\mathrm{,3}}\)

by noting (\({T}_{h,i}\)) the set of finite elements having a face contained in the \({\Gamma }_{i}\) border.

  • With the same logic, all the faces of the triangulation (\({T}_{h}\)) are broken down in the form

\({S}_{h}:={S}_{h,O}\cup {S}_{h\mathrm{,1}}\cup {S}_{h\mathrm{,2}}\cup {S}_{h\mathrm{,3}}\)

with

\(\forall i\in \left\{\mathrm{1,2}\mathrm{,3}\right\}{S}_{h,i}:=\left\{\partial K/K\in {T}_{h}\partial K\subset {\mathrm{\Gamma }}_{i}\right\}=\underset{K\in {T}_{h,i}}{\cup S(K)}\)

Likewise, all the nodes of the triangulation (*h) break down in the form

\({N}_{h}:={N}_{h,O}\cup {N}_{h\mathrm{,1}}\cup {N}_{h\mathrm{,2}}\cup {N}_{h\mathrm{,3}}\)

  • The « bubble » function associated with \(K\) (resp. \(F\)) is noted \({\psi }_{K}\) (resp. \({\psi }_{F}\)).

Note:

It is the function of \(D(\Omega )\) (set of indefinitely differentiable and compact supported functions) resulting from the truncation theorem on a compact: its support is limited to the compact in question (here \(K\) or \(F\) ) and it is worth between 0 and 1 on its inner (in the topological sense of the term). It is therefore zero on the border of the compact and outside of the compact.

  • We note \({P}_{F}\) the bearing operator on \(K\) and traces on \(F\), built from a bearing operator fixed to the reference element.

  • The union of the finite elements of the triangulation sharing at least one face with \(K\) is noted

\({\Delta }_{K}:=\underset{S(K)\cap S(K\text{'})\ne \varnothing }{\cup K\text{'}}\)

The union of the finite elements of the triangulation containing*F in their border is noted

\({\Delta }_{F}:=\underset{F\in S(K\text{'})}{\cup K\text{'}}\)

  • The union of the finite elements of triangulation that share at least one node with \(K\) (resp. with \(F\)) is noted

\({\omega }_{K}:=\underset{N(K)\cap N(K\text{'})\ne \varnothing }{\cup K\text{'}}\) (\({\omega }_{F}:=\underset{N(F)\cap N(K\text{'})\ne \varnothing }{\cup K\text{'}}\) respectively).

_images/Object_266.svg

Figure 4.1-a: Designation of neighborhood types for Ket F.

4.2. Increase in the global spatial error#

So let’s see how to get a local error indicator that can be calculated from the data and the discrete solution \({({u}_{h}^{n})}_{n}\). As the discretized workspace is included in the continuous space \({V}_{h}\subset V\), we can reuse [éq 3.2-3] with \({v}_{h}\). By subtracting [éq 3.4-10] from it it happens (to \(n\) and \(h\) fixed and assuming (\({H}_{6}\)) and (\({H}_{7}\)))

\({(\rho {C}_{p}({u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}),{\nu }_{h})}_{\mathrm{0,}\Omega }\text{+}\Delta ta(({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1}),{\nu }_{h})={(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),{\nu }_{h})}_{\mathrm{0,}\Omega }(\forall {\nu }_{h}\in {V}_{h})\) eq 4.2-1

Notes:

  • This relationship states the orthogonal nature of the spatial error with respect to the elements of Vh.

  • It also assumes that the discretization is «  **consistent », that is to say that there are no* **additional errors introduced by the numerical integration of integrals. In practice this is of course not the case!*

Consider the following linear form

\(A(\nu ):={(\rho {C}_{p}({u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}),\nu )}_{\mathrm{0,}\Omega }\text{+}\Delta ta({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1},\nu )(\forall \nu \in V)\) eq 4.2-2

which will serve as a common thread during this demonstration. By expanding it via [éq 4.2-1], we get

\(\begin{array}{}A(\nu )={(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),\nu )}_{\mathrm{0,}\Omega }\text{+}{(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),({\nu }_{h}\text{-}\nu ))}_{\mathrm{0,}\Omega }\text{+}\\ (\forall \nu \in V){(\rho {C}_{p}({u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}),(\nu \text{-}{\nu }_{h}))}_{\mathrm{0,}\Omega }\text{+}\Delta ta(({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1}),\nu \text{-}{\nu }_{h})\end{array}\) eq 4.2-3

By taking [éq 3.2-3] after replacing \({v}_{h}\) with \(v-{v}_{h}\in V\), we can build

\(\begin{array}{}{(\rho {C}_{p}({u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}\text{-}{u}^{n}\text{+}{u}_{h}^{n}),\nu \text{-}{\nu }_{h})}_{\mathrm{0,}\Omega }\text{+}\Delta ta{({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1},\nu \text{-}{\nu }_{h})}_{\mathrm{0,}\Omega }=\\ (\forall \nu \in V)\Delta t{({b}_{\theta }^{n\text{+}1},\nu \text{-}{\nu }_{h})}_{\mathrm{0,}\Omega }\text{-}\Delta ta({u}_{\theta ,h}^{n\text{+}1},\nu \text{-}{\nu }_{h})\text{-}{(\rho {C}_{p}({u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}),\nu \text{-}{\nu }_{h})}_{\mathrm{0,}\Omega }\end{array}\) eq 4.2-4

So \(A(v)\) becomes

\(\begin{array}{}A(\nu )={(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),\nu )}_{\mathrm{0,}\Omega }\text{+}\Delta t({b}_{\theta }^{n\text{+}1},\nu \text{-}{\nu }_{h})\text{-}\\ (\forall \nu \in V){(\rho {C}_{p}({u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}),\nu \text{-}{\nu }_{h})}_{\mathrm{0,}\Omega }\text{-}\Delta ta({u}_{\theta ,h}^{n\text{+}1},\nu \text{-}{\nu }_{h})\end{array}\) eq 4.2-5

Then we break down the last three terms on each element \(K\) of the triangulation and we apply Green’s formula to the last

\(\begin{array}{}A(\nu )={(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),v)}_{\mathrm{0,}\Omega }\text{+}\Delta t\sum _{K\in {T}_{h}}\underset{K}{\int }({\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{\theta ,h}^{n\text{+}1}))(\nu \text{-}{\nu }_{h})\text{dx}\\ \text{-}\frac{\Delta t}{2}\sum _{F\in {S}_{h,\Omega }}\underset{F}{\int }\left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right](\nu \text{-}{\nu }_{h})d\sigma \\ \forall \nu \text{-}{\nu }_{h}\in V\text{+}\Delta t\sum _{F\in {S}_{h\mathrm{,2}}}\underset{F}{\int }({\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n})(\nu \text{-}{\nu }_{h})d\sigma \\ \text{+}\Delta t\sum _{F\in {S}_{h\mathrm{,3}}}\underset{F}{\int }({\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1})(\nu \text{-}{\nu }_{h})d\sigma \end{array}\) eq 4.2-6

note

We allowed ourselves to replace the duality brackets of [:ref:`éq 3.2-4 <éq 3.2-4>`] by integrals and we can apply Green’s formula because on the compact :math:`K` the hypotheses ( \({H}_{4}\)) and (\({H}_{5}\)) are true (by replacing \(\Omega\) by \(K\) and \({\Gamma }_{i}\) by \(\partial K\cap {\Gamma }_{i}\)). So we have

\(\nu-{\nu}_{h}\in {H}^{1}(K),{u}_{h}\in {H}^{2}(K),\stackrel{ˆ}{s}\in {L}^{2}(K),\stackrel{ˆ}{g}\in {L}^{2}(\partial K\cap {\Gamma }_{2})\text{et}\stackrel{ˆ}{h}\in {L}^{2}(\partial K\cap {\Gamma }_{3})\)

Recall some properties of the \({L}^{2}\) -local projection operator \({\Pi }_{h}\) introduced by P. CLEMENT [bib8]

\(\begin{array}{}{\Pi }_{h}:V\subset {L}^{2}(\Omega)\to {V}_{h}\\ \nu\to {\nu}_{h}\end{array}\) eq 4.2-8

In particular, it checks for increases in projection errors

\(\begin{array}{}\forall \nu\in V{\parallel \nu-{\Pi }_{h}\nu\parallel }_{\mathrm{0,}K}:={\parallel \nu-{\nu}_{h}\parallel }_{\mathrm{0,}K}\le {C}_{4}{h}_{K}{\parallel \nu\parallel }_{\mathrm{1,}{\omega }_{K}}\\ \forall K\in {T}_{h},\forall F\in S(K){\parallel \nu-{\Pi }_{h}\nu\parallel }_{\mathrm{0,}F}:={\parallel \nu-{\nu}_{h}\parallel }_{\mathrm{0,}F}\le {C}_{5}\sqrt{{h}_{F}}{\parallel \nu\parallel }_{\mathrm{1,}{\omega }_{F}}\end{array}\) eq 4.2-9

where the constants \({C}_{4}\) and \({C}_{5}\) depend on the smallest angles of the triangulation. By taking this spatial projection operator and applying the Cauchy-Schwartz inequality to [éq 4.2-6] it therefore happens:

\(\begin{array}{}A(\nu )\text{-}{(\rho {C}_{p}({u}^{n}\text{-}{u}_{h}^{n}),\nu )}_{\mathrm{0,}\Omega }\le \Delta {\text{tC}}_{4}\sum _{K\in {T}_{h}}{h}_{K}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}K}{\parallel \nu \parallel }_{\mathrm{1,}{\omega }_{K}}\\ \text{+}\frac{\Delta t}{2}{C}_{5}\sum _{F\in {S}_{h,\Omega }}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{\mathrm{0,}F}{\parallel \nu \parallel }_{\mathrm{1,}{\omega }_{F}}\\ \forall \nu \in V\text{+}\Delta {\text{tC}}_{5}\sum _{F\in {S}_{h\mathrm{,2}}}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}{\parallel \nu \parallel }_{\mathrm{1,}{\omega }_{F}}\\ \text{+}\Delta {\text{tC}}_{5}\sum _{F\in {S}_{h\mathrm{,3}}}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}F}{\parallel \nu \parallel }_{\mathrm{1,}{\omega }_{F}}\end{array}\) eq 4.2-10

This inequality clearly reveals a possible formulation of the pure residue indicator:

Definition 10

As part of the Code_Aster linear transient thermal operator, the suit \({({\eta }^{n}(K))}_{0\le n\le N}^{K\in {T}_{h}}\) of theoretical local indicators can be written in the form

\(\begin{array}{}{\eta }^{n\text{+}1}(K):={h}_{K}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{h,\theta }^{n\text{+}1})\parallel }_{\mathrm{0,}K}\text{+}\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{\mathrm{0,}F}\text{+}\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}\text{+}\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}F}\end{array}\) eq 4.2-11

It is initialized by

\(\begin{array}{}{\eta }^{0}(K):={h}_{K}{\parallel {\stackrel{ˆ}{s}}^{0}\text{+}\text{div}(\lambda \nabla {u}_{h}^{0})\parallel }_{\mathrm{0,}K}\text{+}\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {u}_{h}^{0}}{\partial n}\right]\parallel }_{\mathrm{0,}F}\text{+}\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{^}{g}}^{0}\text{-}\lambda \frac{\partial {u}_{h}^{0}}{\partial n}\parallel }_{\mathrm{0,}F}\text{+}\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{h}}^{0}\text{-}\lambda \frac{\partial {u}_{h}^{0}}{\partial n}\text{-}{h}^{0}{u}_{h}^{0}\parallel }_{\mathrm{0,}F}\end{array}\) eq 4.2-12

The suit \({({\mathrm{\eta }}^{n}(O))}_{0\le n\le N}\) of theoretical global indicators is defined as

\(\forall 0\le n\le N{\eta }^{n}(\Omega):={(\sum _{K\in {T}_{h}}{\eta }^{n}{(K)}^{2})}^{\frac{1}{2}}\) eq 4.2-13

Notes:

By placing ourselves in the particular framework [:ref:`éq 3.1-9 <éq 3.1-9>`] of the article [:ref:`bib6 <bib6>`] with an implicit schema (:math:`theta =1`) we find the definition (24) pp432. * * Regardless of the initialization used for the thermal calculation, we start the time series of mapping error indicators as if we were stationary: no term with a finite time difference, \(n+1=0\) (in the Code_Aster a transient temperature field is initialized at the index 0) and \(\theta =1\) . * It should be noted that this indicator is composed of four terms: the main term , called volume error term , controlling the correct verification of the heat equation, to which are added three secondary terms verifying the correct compliance of the limit conditions ( terms of jump, flow and exchange) ). En 2D- PLAN * ** or en* 3D * (resp. en 2D- AXI ), if the unit of geometry is the meter, the unit of the first is the W.m * (resp. 3D* ** (resp. 3D* **) (resp.*3D* * **) (resp.*3D* * ) (resp. 3D* **) (resp. en*:math:Wtext{.}mtext{.}{text{rad}}^{-1}`*), if the unit of geometry is the meter, the unit of the former is the* *W.m* * *(resp.* MD* **) (resp.* *) and that of the other terms is* :math:`Wtext{.}{m}^{frac{1}{2}} (resp. \(W\text{.}{m}^{\frac{1}{2}}\text{.}{\text{rad}}^{-1}\) ). So pay attention to the units taken into account for geometry when interested in the raw value of the indicator and not in its relative value! * Inspired by the increases developed by R. VERFURTH (cf. [bib7] pp84-94) for the Poisson equation, we could have taken as an indicator the root of the sum of the squares of the terms mentioned above. \({\tilde{\eta }}^{n\text{+}1}(K):={\left\{\begin{array}{}{h}_{K}^{2}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{h,\theta }^{n\text{+}1})\parallel }_{\mathrm{0,}K}^{2}\text{+}\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}{h}_{F}{\parallel \left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{\mathrm{0,}F}^{2}\text{+}\\ \sum _{F\in {S}_{2}(K)}{h}_{F}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}^{2}\text{+}\sum _{F\in {S}_{3}(K)}{h}_{F}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\end{array}\right\}}^{\frac{1}{2}}\) eq 4.2-14 This definition leads to an increase in the overall error that is more optimal than that which will be identified later. But in order to remain consistent with the writings of B. METIVET [bib6] and with the linear mechanics estimator already implemented in the code, we preferred to stick to the definition version 10.

Based on [éq 4.2-10] and definition 10 we can then exhume the following increase in the global error:

Property 11

Under the assumptions of properties 6, of (\({H}_{6}\)) and using definition 10, we have, at the global level, the small « increase in the error (with \({K}_{2}({\parallel \lambda \parallel }_{\infty ,\Omega},P(\Omega),{C}_{4},{C}_{5})>0\)) via the history of the indicators

\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({u}^{n}\text{-}{u}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}4(1\text{-}\theta )\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\rho {C}_{p}}({u}^{m}\text{-}{u}_{h}^{m})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{m\text{+}1}\text{-}{u}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le (4\theta \text{-}3){\parallel \sqrt{\rho {C}_{p}}({u}_{0}\text{-}{u}_{h}^{0})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}{K}_{2}\Delta t\sum _{m=0}^{n}{({\eta }^{m}(\Omega ))}^{2}\\ \end{array}\) eq 4.2-15

or more simply

\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({u}^{n}\text{-}{u}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{m\text{+}1}\text{-}{u}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le {\parallel \sqrt{\rho {C}_{p}}({u}_{0}\text{-}{u}_{h}^{0})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}{K}_{2}\Delta t\sum _{m=0}^{n}{({\eta }^{m}(\Omega ))}^{2}\\ \end{array}\) eq 4.2-16

Proof:

The [éq 4.2-15] [éq 4.2-16] estimates are obtained by repeating the same process as for properties 5, 6, and 7. In [éq 4.2-10] we take the particular test function

\(\nu :={u}_{\theta }^{n\text{+}1}-{u}_{\theta ,h}^{n\text{+}1}\) eq 4.2-17

We rule out the exchange term by the usual argument

\(\underset{{\Gamma }_{3}}{\int }{(h(u-{u}_{h}))}_{\theta }^{n\text{+}1}({u}_{\theta }^{n\text{+}1}-{u}_{\theta ,h}^{n\text{+}1})\text{dx}>0\) eq 4.2-18

The trick [éq 3.1-4] must be applied to the \((2\theta -1)\underset{\Omega }{\int }\rho {C}_{p}({u}^{n\text{+}1}-{u}_{h}^{n\text{+}1})({u}^{n}-{u}_{h}^{n})\text{dx}\) cross term and to the product involving the indicator. We then have to find the parameters \(\alpha\) and \(\beta\) verifying a system of the type [éq 3.2-8]

\(\begin{array}{}\mid 2-\frac{{P}^{2}(\Omega)}{{\parallel \lambda \parallel }_{\infty ,\Omega}}{\alpha }^{2}=1\\ \mid \mathrm{2\theta }-{\beta }^{2}\mid 1-\mathrm{2\theta }\mid =1\end{array}\) eq 4.2-19

which only admits a solution if the schema is unconditionally stable (\(\mathrm{\theta }\ge \frac{1}{2}\)). Hence the increase [éq 4.2-15] [éq 4.2-16] by taking

\({K}_{2}=\frac{{P}^{2}(\Omega)}{{\parallel \lambda \parallel }_{\infty ,\Omega}}\text{max}({C}_{4}^{2},{C}_{5}^{2})\) eq 4.2-20

The « coarser » inequality [éq 4.2-16] results from the same argument as for corollary 7.

notes

  • By placing ourselves in the particular framework [éq 3.1-9] of the article [bib6] with an implicit schema (\(\theta =1\)) we find the inequality (25) pp432 (with \(c=\mathrm{max}(\mathrm{1,}K2)\)).

  • By taking the less restrictive assumptions (\({H}_{4}\)) and (\({H}_{5}\)), we find a « strong » version of this property.

  • This property can be demonstrated more quickly by noticing that the inequality [éq 4.2-10] is similar to the equation for the semi-discretized problem in time [éq 3.2-3]: except for the inequality, by changing \(u\) by \(u-{u}_{h}\) and taking the second member of [éq 4.2-10] as a term \(({b}_{\theta }^{n},v)\) and taking the second member of [] as the term. Corollary 7 can then be directly applied to it, which is the equivalent of the estimate sought!

From [:ref:`éq 4.2-15 <éq 4.2-15>`] [:ref:`éq 4.2-16 <éq 4.2-16>`]**it appears that, at a given moment, the error in approximating the Cauchy condition and the history of the global indicators affects the overall quality of the solution obtained. It is therefore possible to globally minimize the approximation error due to finite elements over time by remelining the structure « wisely », via a series of indicators. Because, in practice, we see that the refinement of the meshes makes it possible to reduce their error and therefore to reduce the temporal sum of the indicators. The global error will (and this is moral) hit the floor value of the error in approximating the initial condition (which will also tend to decrease, of course!). The indicator « overestimates » the spatial error overall. *

  • With the other indicator variant [éq 4.2-14] we find the same type of increase. However the constant \({K}_{2}\) is changing. It is found multiplied by the constant \({C}_{6}\) verifying (cf. [bib7] pp90)

    \(\sum _{K\in {T}_{h}}{\parallel \nu\parallel }_{\mathrm{1,}{\omega }_{K}}^{2}+\sum _{F\in {S}_{h}}{\parallel \nu\parallel }_{\mathrm{1,}{\omega }_{F}}^{2}\le {C}_{6}{\parallel \nu\parallel }_{\mathrm{1,}\Omega}^{2}\) eq 4.2-21

    \({\tilde{K}}_{2}:={C}_{6}{K}_{2}\) eq 4.2-22

According to the definitions [éq 2-1-8], [éq 2-1-10] to [éq 2-1-13] whether the consideration of Dirichlet boundary conditions (generalized or not), via the Lagrange ddls, is accurate (which is the case in Code_Aster)

\[\]

: label: EQ-None

forall h {text {Rf}} _ {h}} ^ {n} := {Pi} _ {h} {text {Rf}} ^ {n} = {text {Rf}}} ^ {n}} ^ {n} =text {n}} =text {Rf} =text {Rf} (nDeltat) 0le nle N

the previous property then produces the following corollary:

Corollary 11bis

Under the assumptions of property 11 assuming (H8), we have the increase of the global spatial error expressed in temperature

\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({T}^{n}-{T}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}+\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({T}_{\theta }^{m\text{+}1}-{T}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le {\parallel \sqrt{\rho {C}_{p}}({T}_{0}-{T}_{h}^{0})\parallel }_{\mathrm{0,}\Omega }^{2}+{K}_{2}\Delta t\sum _{m=0}^{n}{({\eta }^{m}(\Omega ))}^{2}\\ \end{array}\) eq 4.2-23

using the definition 10 of the indicator also expressed in temperature

\(u\Rightarrow T,\stackrel{ˆ}{s}\Rightarrow s,\stackrel{ˆ}{g}\Rightarrow g\text{et}\stackrel{ˆ}{h}\Rightarrow {\text{hT}}_{\text{ext}}\) eq 4.2-24

4.3. Different types of indicators possible#

Extrapolating a remark from [bib5] (pp194-195) it appears that the increases in property 11 can be exhumed by taking as an indicator

\(\begin{array}{}{\eta }_{p,t}^{n\text{+}1}(K):={h}_{K}^{r}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{h,\theta }^{n\text{+}1})\parallel }_{{L}^{p}(K)}\text{+}\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}{h}_{F}^{s}{\parallel \left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{{L}^{t}(K)}\text{+}\\ \sum _{F\in {S}_{2}(K)}{h}_{F}^{s}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{{L}^{t}(K)}\text{+}\sum _{F\in {S}_{3}(K)}{h}_{F}^{s}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1}\parallel }_{{L}^{t}(K)}\end{array}\) eq 4.3-1

where the constants \(r\) and \(s\) are equal

\(\begin{array}{}t\ge \mathrm{1,}p>1(\text{2D}q=2)\text{ou}p\ge \frac{6}{5}(\text{3D}q=3)\\ \mid \begin{array}{c}r(q,p):=q+1-\frac{q}{2}-\frac{q}{p}\\ s(q,t):=q-\frac{1}{2}-\frac{q-1}{2}-\frac{q-1}{t}\end{array}\end{array}\) eq 4.3-2

Note:

Just to introduce this generic form of indicators, we go from the Hilbertian notation of space norms to the Lebesgue notation

It is parameterized by the types of volume and surface standards that are used to obtain it. Contrary to the indicator we have chosen (\({\eta }_{\mathrm{2,2}}^{n\text{+}1}(K)\) which corresponds to \(p=t=2\)), some use the volume norm \({L}^{1}\) (\(p=t=2\)) or on the contrary the infinite norm.

This last formulation, like its simplified form of definition 10 (or [éq 4.2-14]), is indeed an indicator of an a posteriori error because its calculation only requires knowledge of the materials, loads, geometric data, \(\theta\), and and the approximate solution field \({u}_{h}\) of the thermal problem in question. However, the exact estimation of the indicator is not always possible when there are complicated loads. Two approaches are then possible:

Or we**approximate the integrals**that enter into the composition of definition 10 by a**quadrature formula. *

Or**we approximate the loads* by a linear combination of simpler functions that can allow exact integration. In general, the same architecture as that which was put in place for the finite elements modeling the temperature field is used.

Note:

  • In both cases the loads are « prisoners of the finite element vision » chosen to model the solution field.

These two strategies are equivalent and in code_asterIt was the first that was retained: the volume integral is calculated by a Gauss formula, the surface ones by a Newton-Cotes formula.

Both introduce a bias in the estimator calculation that can be represented by introducing the approximate versions of the loads and the source (in the initial problem in \(T\) and in the transformed problem in \(u\))

\({s}_{\theta ,h}^{n\text{+}1},{g}_{\theta ,h}^{n\text{+}1},{T}_{\text{ext},\theta ,h}^{n\text{+}1}\text{et}{h}_{\theta ,h}^{n\text{+}1}\) eq 4.3-3

\({\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1},{\stackrel{ˆ}{g}}_{\theta ,h}^{n\text{+}1},{\stackrel{ˆ}{h}}_{\theta ,h}^{n\text{+}1}\text{et}{h}_{\theta ,h}^{n\text{+}1}\) eq 4.3-4

in the volume (for the source) and surface approximation spaces (for the loads)

\(\begin{array}{}{X}_{h}(\Omega):=\left\{{\nu}_{h}\in {L}^{2}(\Omega)/\forall K\in {T}_{h}{\nu}_{{h}_{\mid K}}\in {P}_{{l}_{1}}(K)\right\}\\ {X}_{h}({\mathrm{\Gamma }}_{i}):=\left\{{\nu}_{h}\in {L}^{2}({\mathrm{\Gamma }}_{i})/\forall F\in {S}_{h,i}{\nu}_{{h}_{\mid F\cap {\mathrm{\Gamma }}_{i}}}\in {P}_{{l}_{i}}(F\cap {\mathrm{\Gamma }}_{i})\right\}\end{array}\) eq 4.3-5

In fact, two types of numerical errors are introduced when calculating the indicator: the one inherent in quadrature formulas (for high-order polynomial loads) and the one due to the volume term. In fact, the latter requires a double derivation that is carried out in three steps because in Code_Aster the use of second derivatives of form functions is not recommended.

Note:

They have recently been introduced to deal with the derivation of the energy return rate (cf. [R7.02.01 §Annex 1]).

On the one hand, we calculate (in the thermal operator) the heat flow at the Gauss points, then we extrapolate the values to the corresponding nodes by local smoothing (cf. [R3.06.03] CALC_CHAMP with THERMIQUE =” FLUX_ELNO “) before calculating the divergence of the flow vector at the Gauss points. With quadratic finite elements the intermediate operation is only approximate (we assign the value to the median nodes the half-sum of their values at the extreme nodes). However, numerical tests (limited) have shown that, even in \({P}_{2}\), this approach does not provide results that are very different from those obtained by a direct calculation using the correct second derivatives.

Note:

  • The indices \({l}_{1}\) , \({l}_{2}\) , , ,, :math:`{l}_{3}` of these polynomial spaces may be any and different from that of the approximate solution: :math:`k`. However, to prevent these terms from becoming predominant (* **it is a question of estimating the error on the solution rather than on the modeling of the loads) we will tend to take* \({l}_{i}\ge k-2(i=\mathrm{1,2}\mathrm{,3})\) .

The definition 10 and the associated low estimate 11 are then rewritten in the following form. This new definition, \({\eta }_{R}^{n+1}(K)\), is indexed by a \(R\) ( we use the usual notations of [bib6] and [bib7]) (for « real ») (for « real ») in order to clearly indicate that it corresponds better to the values that are actually calculated in the code.

Definition 12

As part of the Code_Aster linear transient thermal operator, the suit \({({\eta }_{R}^{n}(K))}_{0\le n\le N}^{K\in {T}_{h}}\) of real local indicators can be written in the form

\(\begin{array}{}{\eta }_{R}^{n\text{+}1}(K):={h}_{K}{\parallel {\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1}\text{-}\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}\text{-}{u}_{h}^{n}}{\Delta t}\text{+}\text{div}(\lambda \nabla {u}_{h,\theta }^{n\text{+}1})\parallel }_{\mathrm{0,}K}\text{+}\frac{1}{2}\sum _{F\in {S}_{\Omega }(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\parallel }_{\mathrm{0,}F}\text{+}\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{g}}_{\theta ,h}^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\parallel }_{\mathrm{0,}F}\text{+}\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{h}}_{\theta ,h}^{n\text{+}1}\text{-}\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\text{-}{({h}_{h}{u}_{h})}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}F}\end{array}\) eq 4.3-6

It is initialized by

\(\begin{array}{}{\eta }_{R}^{0}(K):={h}_{K}{\parallel {\stackrel{ˆ}{s}}_{h}^{0}+\text{div}(\lambda \nabla {u}_{h}^{0})\parallel }_{\mathrm{0,}K}+\frac{1}{2}\sum _{F\in {S}_{\Omega}(K)}\sqrt{{h}_{F}}{\parallel \left[\lambda \frac{\partial {u}_{h}^{0}}{\partial n}\right]\parallel }_{\mathrm{0,}F}+\\ \sum _{F\in {S}_{2}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{g}}_{h}^{0}-\lambda \frac{\partial {u}_{h}^{0}}{\partial n}\parallel }_{\mathrm{0,}F}+\sum _{F\in {S}_{3}(K)}\sqrt{{h}_{F}}{\parallel {\stackrel{ˆ}{h}}_{h}^{0}-\lambda \frac{\partial {u}_{h}^{0}}{\partial n}-{h}_{h}^{0}{u}_{h}^{0}\parallel }_{\mathrm{0,}F}\end{array}\) eq 4.3-7

The suit \({({\eta }^{n}(\Omega))}_{0\le n\le N}\) of real global indicators is defined as

\(\forall 0\le n\le N{\eta }_{R}^{n}(\Omega):={(\sum _{K\in {T}_{h}}{\eta }_{R}^{n}{(K)}^{2})}^{\frac{1}{2}}\) eq 4.3-8

Note:

You can make the same remarks as for your « theoretical » alter ego. They also come in accordance with the formulations [:ref:`éq 4.2-14 <éq 4.2-14>`] :math:`{tilde{eta }}_{R}^{n}(K)` and [:ref:`éq 4.3-1 <éq 4.3-1>`], [:ref:`éq 4.3-2 <éq 4.3-2>`] :math:`{eta }_{R,p,t}^{n}(K)` . *

Based on the results of property 11, definition 12 and triangular inequality, we can then exhume the following increase in the real global error (we only used the simplified version):

Property 13

Under the assumptions of property 6, of (H6) and using definition 12, we have, at the global level, the small « increase in the error (with \({K}_{2}({\parallel \lambda \parallel }_{\infty ,\Omega},P(\Omega),{C}_{4},{C}_{5})>0\)) via the history of real indicators

\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({u}^{n}\text{-}{u}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{m\text{+}1}\text{-}{u}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le {\parallel \sqrt{\rho {C}_{p}}({u}_{0}\text{-}{u}_{h}^{0})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}{K}_{2}\Delta t\sum _{K\in {T}_{h}}{({\eta }_{R}^{0}(K))}^{2}\text{+}\sum _{m=0}^{n\text{-}1}\left\{{({\eta }_{R}^{m\text{+}1}(K))}^{2}\text{+}{h}_{K}^{2}{\parallel {\stackrel{ˆ}{s}}_{\theta ,h}^{m\text{+}1}\text{-}{\stackrel{ˆ}{s}}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}K}^{2}\right\}\text{+}\\ {K}_{2}\Delta t\sum _{K\in {T}_{h}}\sum _{m=0}^{n\text{-}1}\left\{\sum _{F\in {S}_{2}(K)}{h}_{F}{\parallel {\stackrel{ˆ}{g}}_{\theta ,h}^{m\text{+}1}\text{-}{\stackrel{ˆ}{g}}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\text{+}\sum _{F\in {S}_{3}(K)}{h}_{F}{\parallel {\stackrel{ˆ}{h}}_{\theta ,h}^{m\text{+}1}\text{-}{\stackrel{ˆ}{h}}_{\theta }^{m\text{+}1}\text{-}{({h}_{h}{u}_{h})}_{\theta }^{m\text{+}1}\text{+}{({\text{hu}}_{h})}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\right\}\\ \end{array}\) eq 4.3-9

Under (H8), we have the same expression in temperature

\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}({T}^{n}\text{-}{T}_{h}^{n})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla ({T}_{\theta }^{m\text{+}1}\text{-}{T}_{\theta ,h}^{m\text{+}1})\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\le {\parallel \sqrt{\rho {C}_{p}}({T}_{0}\text{-}{T}_{h}^{0})\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}{K}_{2}\Delta t\sum _{K\in {T}_{h}}{({\eta }_{R}^{0}(K))}^{2}\text{+}\sum _{m=0}^{n\text{-}1}\left\{{({\eta }_{R}^{m\text{+}1}(K))}^{2}\text{+}{h}_{K}^{2}{\parallel {s}_{\theta ,h}^{m\text{+}1}\text{-}{s}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}K}^{2}\right\}\text{+}\\ {K}_{2}\Delta t\sum _{K\in {T}_{h}}\sum _{m=0}^{n\text{-}1}\left\{\sum _{F\in {S}_{2}(K)}{h}_{F}{\parallel {g}_{\theta ,h}^{m\text{+}1}\text{-}{g}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\text{+}\sum _{F\in {S}_{3}(K)}{h}_{F}{\parallel {({h}_{h}({T}_{\text{ext},h}\text{-}{T}_{h}))}_{\theta }^{m\text{+}1}\text{-}{(h({T}_{\text{ext}}\text{-}{T}_{h}))}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}F}^{2}\right\}\\ \end{array}\) eq 4.3-10

using definition 12 of the indicator also expressed in temperature

\(u\Rightarrow T,\stackrel{ˆ}{s}\Rightarrow s,\stackrel{ˆ}{g}\Rightarrow g\text{et}\stackrel{ˆ}{h}\Rightarrow {\text{hT}}_{\text{ext}}\) eq 4.3-11

Note:

  • As with theoretical value, there is a moral to the story because, when we refine, the global error will come up against the floor value due to approximations of the initial condition, the boundary conditions and the source. You can’t get better quality results than the input data for the problem!

4.4. Reduction of the local spatial error#

Before exhuming the reduction of the spatial error, we will have to introduce some additional results:

Lemma 14

We show that there are strictly positive constants C i (i=6… 11) verifying

\(\begin{array}{}\forall \nu \in {P}_{\text{sup}\left\{k,{l}_{1},{l}_{2},{l}_{3}\right\}}(K){C}_{6}{\parallel {\psi }_{K}\nu \parallel }_{\mathrm{0,}K}\le {\parallel \nu \parallel }_{\mathrm{0,}K}\le {C}_{7}{\parallel {\psi }_{{K}^{\frac{1}{2}}}\nu \parallel }_{\mathrm{0,}K}\\ {\parallel \nabla {\psi }_{K}\nu \parallel }_{\mathrm{0,}K}\le {C}_{8}{h}_{K}^{\text{-}1}{\parallel {\psi }_{K}\nu \parallel }_{\mathrm{0,}K}\\ \forall \nu \in {P}_{\text{sup}\left\{k,{l}_{1},{l}_{2},{l}_{3}\right\}}(F){C}_{9}{h}_{F}^{\text{-}\frac{1}{2}}{\parallel {\psi }_{K}{P}_{F}\nu \parallel }_{\mathrm{0,}{\Delta }_{F}}\le {\parallel \nu \parallel }_{\mathrm{0,}F}\le {C}_{\text{10}}{\parallel {\psi }_{{F}^{\frac{1}{2}}}\nu \parallel }_{\mathrm{0,}F}\\ {\parallel \nabla {\psi }_{K}\nu \parallel }_{\mathrm{0,}{\Delta }_{F}}\le {C}_{\text{11}}{h}_{F}^{\text{-}1}{\parallel {\psi }_{K}\nu \parallel }_{\mathrm{0,}{\Delta }_{F}}\\ \end{array}\) eq 4.4-1

Proof:

We move on to the reference element then we use the fact that the norms are equivalent on the polynomial spaces in question, because they are finite in dimension (cf. [bib5] pp196-98, [bib7] [§1], [], []).

These preliminary results are crucial to determine a reduction in the local error by the real indicator. But we’ll see that we can only get a local inverse of [éq 4.3-9], [éq 4.3-10].

Property 15

Under the assumptions of property 6, of (\({H}_{6}\)) and based on definition 12 and lemma 14, we have, at the local level, the « low » reduction of the error (with \({K}_{3}({C}_{i},i=6\cdots \text{11})>0\)) (with)**via the real indicator

\(\begin{array}{}{\eta }_{R}^{n\text{+}1}(K)\le {K}_{3}\left\{\begin{array}{}{h}_{K}{\parallel \sqrt{\rho {C}_{p}}\frac{{u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}\\ {h}_{K}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}{h}_{F}^{\frac{1}{2}}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{g}}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{2}}\text{+}\\ {h}_{F}^{\frac{1}{2}}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{h}}_{\theta ,h}^{n\text{+}1}\text{-}{({\text{hu}}_{h})}_{\theta }^{n\text{+}1}\text{+}{({h}_{h}{u}_{h})}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{3}}\end{array}\right\}\\ \\ \forall 0\le n\le N\text{-}1\end{array}\) eq 4.4-2

Under (H8), we have the same expression in temperature

\(\begin{array}{}{\eta }_{R}^{n\text{+}1}(K)\le {K}_{3}\left\{\begin{array}{}{h}_{K}{\parallel \sqrt{\rho {C}_{p}}\frac{{T}^{n\text{+}1}\text{-}{T}_{h}^{n\text{+}1}\text{-}{T}^{n}\text{-}{T}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}{\parallel \sqrt{\lambda }\nabla ({T}_{\theta }^{n\text{+}1}\text{-}{T}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}\\ {h}_{K}{\parallel {s}_{\theta }^{n\text{+}1}\text{-}{s}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{+}{h}_{F}^{\frac{1}{2}}{\parallel {g}_{\theta }^{n\text{+}1}\text{-}{g}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{2}}\text{+}\\ {h}_{F}^{\frac{1}{2}}{\parallel {h}_{\theta }^{n\text{+}1}({T}_{\text{ext},\theta }^{n\text{+}1}\text{-}{T}_{\theta }^{n\text{+}1})\text{-}{h}_{\theta ,h}^{n\text{+}1}({T}_{\text{ext},\theta ,h}^{n\text{+}1}\text{-}{T}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{K}\cap {\Gamma }_{3}}\end{array}\right\}\\ \\ \forall 0\le n\le N\text{-}1\end{array}\) eq 4.4-3

using definition 12 of the indicator also expressed in temperature

\(u\Rightarrow T,\stackrel{ˆ}{s}\Rightarrow s,\stackrel{ˆ}{g}\Rightarrow g\text{et}\stackrel{ˆ}{h}\Rightarrow {\text{hT}}_{\text{ext}}\) eq 4.4-4

Proof:

This slightly technical demonstration involves three steps which will consist in increasing each of the terms of the indicator [éq 4.3-6] successively (using the inequalities of property 14) and in combining the increases obtained:

First, we will replace in the equation [éq 4.2-6] the term in \(\nu-{\nu}_{h}\) by the product \({w}_{K}\) involving the « bubble » function of \(K\)

\(\begin{array}{}\forall K\in {T}_{h}{\nu }_{K}:={\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1}-\rho {C}_{p}\frac{{u}_{h}^{n\text{+}1}-{u}_{h}^{n}}{\Delta t}+\text{div}(\lambda \nabla {u}_{\theta ,h}^{n\text{+}1})\\ {w}_{K}:={\psi }_{K}{\nu }_{K}\end{array}\) eq 4.4-5

Hence the succession of increases, via [éq 4.4-1] and the Cauchy-Schwartz inequality,

\(\begin{array}{}{\parallel {\nu }_{K}\parallel }_{\mathrm{0,}K}^{2}\le {C}_{7}^{2}\underset{K}{\int }{w}_{K}{\nu }_{K}\text{dx}\le {C}_{7}^{2}\left\{{(\frac{{u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t},{w}_{K})}_{\mathrm{0,}\Omega }\text{+}a({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1},{w}_{K})\text{-}({\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1},{w}_{K})\right\}\\ \le {C}_{7}^{2}\text{max}(\mathrm{1,}{C}_{8})\left\{{\parallel \frac{{u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}K}\text{+}{h}_{K}^{\text{-}1}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}K}\text{+}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}K}\right\}{\parallel {w}_{K}\parallel }_{\mathrm{0,}K}\\ \Rightarrow {\parallel {\nu }_{K}\parallel }_{\mathrm{0,}K}\le \frac{{C}_{7}^{2}}{{C}_{6}}\text{max}(\mathrm{1,}{C}_{8})\left\{{\parallel \frac{{u}^{n\text{+}1}\text{-}{u}_{h}^{n\text{+}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}K}\text{+}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}K}\text{+}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{+}1}\parallel }_{\mathrm{0,}K}\right\}\\ \end{array}\) eq 4.4-6

Then, we repeat the same process for the surface terms \({w}_{F,i}\)

\(\begin{array}{}\forall F\in S(K)\cap {S}_{h,\Omega }{\nu }_{F\mathrm{,1}}:=\left[\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\right]\\ {w}_{F\mathrm{,1}}:={\psi }_{K}{P}_{F}{\nu }_{K\mathrm{,1}}\end{array}\) eq 4.4-7

\(\begin{array}{}\forall F\in S(K)\cap {S}_{h\mathrm{,2}}{\nu }_{F\mathrm{,2}}:={\stackrel{ˆ}{g}}_{h,\theta }^{n\text{+}1}-\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}\\ {w}_{F\mathrm{,2}}:={\psi }_{F}{P}_{F}{\nu }_{F\mathrm{,2}}\end{array}\) eq 4.4-8

\(\begin{array}{}\forall F\in S(K)\cap {S}_{h\mathrm{,3}}{\nu }_{F\mathrm{,3}}:={\stackrel{ˆ}{h}}_{h,\theta }^{n\text{+}1}-\lambda \frac{\partial {u}_{h,\theta }^{n\text{+}1}}{\partial n}-{({h}_{h}{u}_{h})}_{\theta }^{n\text{+}1}\\ {w}_{F\mathrm{,3}}:={\psi }_{F}{P}_{F}{\nu }_{F\mathrm{,3}}\end{array}\) eq 4.4-9

For example, for \(i=1\), the succession of increases, via [éq 4.4-1] and the Cauchy‑Schwartz inequality,

\(\begin{array}{}{\parallel {\nu }_{F\mathrm{,1}}\parallel }_{\mathrm{0,}F}^{2}\le {C}_{\text{10}}^{2}\underset{F}{\int }{w}_{F\mathrm{,1}}{\nu }_{F\mathrm{,1}}d\sigma \le {C}_{\text{10}}^{2}\left\{\begin{array}{}{(\frac{{u}^{n\text{}+\text{}1}\text{-}{u}_{h}^{n\text{}+\text{}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t},{w}_{F\mathrm{,1}})}_{\mathrm{0,}\Omega }\text{+}a({u}_{\theta }^{n\text{}+\text{}1}\text{-}{u}_{\theta ,h}^{n\text{}+\text{}1},{w}_{F\mathrm{,1}})\\ \text{-}{({\stackrel{ˆ}{s}}_{\theta }^{n\text{}+\text{}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{}+\text{}1},{w}_{F\mathrm{,1}})}_{\mathrm{0,}\Omega }\text{-}{({\nu }_{K},{w}_{F\mathrm{,1}})}_{\mathrm{0,}\Omega }\end{array}\right\}\\ \le {C}_{\text{10}}^{2}\text{max}(\mathrm{1,}{C}_{\text{11}})\left\{\begin{array}{}{\parallel \frac{{u}^{n\text{}+\text{}1}\text{-}{u}_{h}^{n\text{}+\text{}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}{\Delta }_{F}}\text{+}{h}_{F}^{\text{}-\text{}1}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{n\text{}+\text{}1}\text{-}{u}_{\theta ,h}^{n\text{}+\text{}1})\parallel }_{\mathrm{0,}{\Delta }_{F}}\\ \text{+}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{}+\text{}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{}+\text{}1}\parallel }_{\mathrm{0,}{\Delta }_{F}}\text{+}{\parallel {\nu }_{K}\parallel }_{\mathrm{0,}{\Delta }_{F}}\end{array}\right\}{\parallel {w}_{F\mathrm{,1}}\parallel }_{\mathrm{0,}{\Delta }_{F}}\\ \Rightarrow {\parallel {\nu }_{F\mathrm{,1}}\parallel }_{\mathrm{0,}F}\le \frac{{C}_{\text{10}}^{2}}{{C}_{9}}\text{max}(\mathrm{1,}{C}_{\text{11}})\left\{\begin{array}{}{h}_{F}^{\frac{1}{2}}{\parallel \frac{{u}^{n\text{}+\text{}1}\text{-}{u}_{h}^{n\text{}+\text{}1}\text{-}{u}^{n}\text{-}{u}_{h}^{n}}{\Delta t}\parallel }_{\mathrm{0,}{\Delta }_{F}}\text{+}{h}_{F}^{\text{}-\text{}\frac{1}{2}}{\parallel \sqrt{\lambda }\nabla ({u}_{\theta }^{n\text{+}1}\text{-}{u}_{\theta ,h}^{n\text{+}1})\parallel }_{\mathrm{0,}{\Delta }_{F}}\\ \text{+}{h}_{F}^{\frac{1}{2}}{\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{}+\text{}1}\text{-}{\stackrel{ˆ}{s}}_{\theta ,h}^{n\text{}+\text{}1}\parallel }_{\mathrm{0,}{\Delta }_{F}}\text{+}{h}_{F}^{\frac{1}{2}}{\parallel {\nu }_{K}\parallel }_{\mathrm{0,}{\Delta }_{F}}\end{array}\right\}\end{array}\) eq 4.4-10

Finally, it is enough to perform the linear combination involving [éq 4.4-9] and [éq 4.4-10] to conclude (because \({h}_{F}\le {h}_{K}\text{et}\forall \nu{\parallel \nu\parallel }_{\mathrm{0,}{\Delta }_{F}}\le {\parallel \nu\parallel }_{\mathrm{0,}{\Delta }_{K}}\text{avec}F\in S(K)\)).

Notes:

This local reduction of the error also occurs according to the formulations [:ref:`éq 4.2-14 <éq 4.2-14>`] :math:`{tilde{eta }}_{R}^{n}(K)` and [:ref:`éq 4.3-1 <éq 4.3-1>`], [:ref:`éq 4.3-2 <éq 4.3-2>`] :math:`{eta }_{R,p,t}^{n}(K)` . *

By placing ourselves in the particular framework [:ref:`éq 3.1-9 <éq 3.1-9>`] of the article [:ref:`bib6 <bib6>`] with an implicit schema (:math:`theta =1`) we find the inequality (29) pp432. *

  • By taking the less restrictive assumptions ( \({H}_{4}\) ) and ( \({H}_{5}\) ), we find a « strong » version of this property.

This result only provides a local inverse of the global increase [:ref:`éq 4.3-9 <éq 4.3-9>`], [:ref:`éq 4.3-10 <éq 4.3-10>`] but within the framework of this type of indicator we will not be able to obtain a better compromise. These estimates are optimal in the sense of [:ref:`bib5 <bib5>`]. They show the equivalence of the Hilbertian sum of the indicators with the spatial part of the global exact error. The equivalence constants are independent of the discretization parameters in space and time, they only depend on the smallest angle of the triangulation. *

  • This increase in the real error indicator shows that if you refine very locally (around \(K\) ) in order to reduce \({\eta }_{R}^{n}(K)\) , you are not guaranteed a decrease in the error in the immediate vicinity of the zone concerned (in \({\Delta }_{K}\) ). The indicator « underestimates » the spatial error locally and only a more macroscopic refinement theoretically reduces the error (cf. property 13) .

4.5. Complements#

The constant \({K}_{3}\), like its previous alter ego, \({K}_{2}\), depends intrinsically on the type of boundary conditions enriching the initial heat equation as well as on the type of temporal and spatial discretization. To try to get rid of this last one constraint, SR. GAGO [bib10] proposes (on a 2D model problem) a dependence of the constant \({K}_{2}\) according to the type of finite elements used. It can be written

\({K}_{2}:=\frac{{\tilde{K}}_{2}}{\sqrt{\text{24}{p}^{2}}}\) eq 4.5-1

where \(p\) is the degree of the interpolation polynomial used (\(p=1\) for TRIA3 and QUAD4, \(p=2\) for TRIA6 and QUAD8 /9). Hence the idea, once the global error indicator has been calculated, to multiply it by this « corrective » constant \(\frac{1}{\sqrt{\text{24}{p}^{2}}}\). This strategy was implicitly adopted for the calculation of the mechanical error indicator (option “ERME_ELEM” from CALC_ERREUR, cf. [R4.10.02 §3]). However, we did not adopt it for thermal studies because this constant was only determined empirically on the 2D Laplace equation. In this way, we do not want to bias the values of the indicators.

Hitherto, there has only been talk of maps of spatial error indicators calculated at a given moment of the calculation transient. But, in fact, there are several ways to build an error indicator on a parabolic problem:

  • we can very well, first of all, semi-discretize the space-intensive formulation and control its spatial error by a posteriori error indicators adapted to the stationary case (in our elliptical case). Then we apply a solver, of varying steps and order, dealing with ordinary differential equations (for example [bib10] [bib11] [bib12]),

  • a second strategy consists in semi-discretizing in time and then in space and in determining the spatial error indicator for a given moment (for example [bib4] [bib6] [bib13]) from the local residues of the semi-discretized form. A linear solver is applied to the variational form making it possible to iteratively build the solution at a given instant from the solution of the previous instant,

  • another possibility consists in simultaneously discretizing time and space on appropriate finite elements and controlling their « space-time » errors in a coupled manner (for example [bib14] [bib15]).

This last scenario is the most appealing from a theoretical point of view because it offers complete control of the error and it makes it possible to avoid unfortunate decoupling as to possible refinements/deraffinations driven by one criterion compared to the other (see next paragraph). However, it is very cumbersome to set up in a large industrial code such as Code_Aster. In fact, in order to be optimal, it assumes nothing less than separate management of the time step by finite elements. From the point of view of the architecture supporting the finite elements of the code, this is a real challenge!

We therefore prefer the second scenario, which has the big advantage of being able to be implemented directly in a code of finite elements, because it relies above all on the resolution of the totally discretized system. It is this type of indicator that has been implemented in N3S, TRIFOU and Code_Aster.

In the context of a true « space-time » discretization of the problem (scenario 3), we obtain, from a point of view, a «  space-time » indicator for each discretization element \(K\times \left[{t}_{n},{t}_{n\text{+}1}\right]\) which is the weighted sum of three terms:

  1. the residue of the calculated solution and the discretized data in relation to the strong formulation of the problem \(({P}_{0})\) evaluated on \(K\times \left[{t}_{n},{t}_{n\text{+}1}\right]\),

  2. the spatial jump through \(\partial K\times \left[{t}_{n},{t}_{n\text{+}1}\right]\) of the associated trace operator (which naturally links the weak and strong formulations via the Green formula),

  3. the time jump through \(K\times \partial \left[{t}_{n},{t}_{n\text{+}1}\right]\) of the calculated solution.

The solution that has been put in place obviously does not allow the term time jump to be explicitly mentioned. It reappears, however,**implicitly, due to the particular temporal semi-discretization method, in**all the terms in**:math:`theta`**from definitions 10 and 12**.

On the other hand, the fact of being mainly interested in spatial discretization and its possible refinement/derefinement should not hide certain contingencies with respect to the management of the time step. In fact, during transient calculations involving sudden variations in loads and/or sources over time, for example thermal shocks, the calculated temperature fields \({T}^{n}(0<n\le N)\) may oscillate spatially and temporally. In addition, they can violate the « principle of the maximum » by taking values outside the limits imposed by the Cauchy condition and the limit conditions. To overcome this parasitic numerical phenomenon, we show, on a canonical case without exchange conditions (cf. [R3.06.07 §2]), that the time step must remain between two limits:

\(\Delta {t}_{\text{min}}(h)<\Deltat <\Delta {t}_{\text{max}}(\mathrm{\theta })\) eq 4.5-2

In practice, it is difficult to have an order of magnitude of these limits, so it is difficult, if one detects oscillations, to modify the time step in order to respect [éq 4.5-2]. On the other hand, this type of operation is not always possible because it is sometimes necessary to take into account precisely the sudden variations in loads (especially when \(\Delta t\) is too small).

When \(\Delta t\) is too big we can operate in Implicit Euler \((\theta =1)\) which will have the effect of erasing the upper bound.

On the other hand when it is too weak, two palliative strategies are available to the user:

  • diagonalize the mass matrix via lumped elements (cf. [R3.06.07 §4] [§5]) proposed in the code (this requires adjustments to treat P 2 elements or 2D_ AXI modeling),

  • reduce the size of the meshes (this increases the computation and memory complexities required).

It is in this perspective that refinements/derefinements practiced based on our indicator may have an effect. Refining will not cause any problem, but by deraffinating you can very well deteriorate the reduction of [éq 4.5-2]. You must therefore be very circumspect if you use the software derefinement option HOMARD (encapsulated for Code_Aster in MACR_ADAP_MAIL option “DERAFFINEMENT” [U7.03.01]) on a test case involving a thermal shock.

We will now summarize the main contributions of the previous theoretical chapters and their ins and outs with respect to the thermal calculation implemented in Code_Aster.