3. Discretization and controllability#
3.1. Controllability of the ongoing problem#
By making no concessions on the assumptions of regularity seen in the previous paragraph, we have the next so-called « low » increase (to use the terminology in force in the article that served as the basis for our study [bib6]).
Property 5
In the abstract variational framework (CVA) defined earlier and assuming that the hypotheses (H1), (H2) and (H3) are true, we have the low controllability of the continuous problem (with \({K}_{1}({\parallel \lambda \parallel }_{\infty ,\Omega },\text{mes}({\Gamma }_{i}),{\gamma }_{\mathrm{0,}i},P(\Omega ))>0\))
\(\begin{array}{}\text{pp}t{\parallel \sqrt{\rho {C}_{p}}u(t)\parallel }_{\mathrm{0,}\Omega }^{2}+\underset{0}{\overset{t}{\int }}{\parallel \sqrt{\lambda }\nabla u(\xi )\parallel }_{\mathrm{0,}\Omega }^{2}d\xi \le {\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}+\\ {K}_{1}\left\{\underset{0}{\overset{t}{\int }}{\parallel \stackrel{ˆ}{s}(\xi )\parallel }_{-1,\Omega }^{2}+{\parallel \stackrel{ˆ}{g}(\xi )\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}^{2}+{\parallel \stackrel{ˆ}{h}(\xi )\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}^{2}d\xi \right\}\end{array}\) eq 3.1-1
Proof:
We are going to detail this slightly technical demonstration here because, on the one hand, specialized literature rarely goes into this level of detail and, on the other hand, we will reuse the same methodology to exhume all the increases that will follow one another in this theoretical part of the document. First of all, by multiplying the equation of [éq 2.1-1] by \(u(t)\), by integrating spatially on \(O\), then temporally on \(\left[\mathrm{0,}t\right]\text{avec}t\in [\mathrm{0,}\tau [\) we obtain, as the material characteristics are supposed to be independent of time,
\(\frac{1}{2}\underset{0}{\overset{t}{\int }}\frac{\partial }{\partial t}{(\rho {C}_{p}u(\xi ),u(\xi ))}_{\mathrm{0,}\Omega }d\xi -\underset{0}{\overset{t}{\int }}{(\text{div}(\lambda \nabla u(\xi )),u(\xi ))}_{\mathrm{0,}\Omega }d\xi =\underset{0}{\overset{t}{\int }}{\langle \stackrel{ˆ}{s}(\xi ),u(\xi )\rangle }_{-1\times \mathrm{1,}\Omega }d\xi\) eq 3.1-2
Using Green’s formula and the boundary conditions of [éq 2.1-1] we get
\(\begin{array}{}\frac{1}{2}({\parallel \sqrt{\rho {C}_{p}}u(t)\parallel }_{\mathrm{0,}\Omega }^{2}-{\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2})+\underset{0}{\overset{t}{\int }}{(\lambda \nabla u(\xi ),\nabla u(\xi ))}_{\mathrm{0,}\Omega }d\xi +\underset{0}{\overset{t}{\int }}h(\xi ){u}^{2}(\xi )d\xi =\\ \underset{0}{\overset{t}{\int }}\left\{{\langle \stackrel{ˆ}{s}(\xi ),u(\xi )\rangle }_{\text{-}1\times \mathrm{1,}\Omega }+{\langle \stackrel{ˆ}{g}(\xi ),u(\xi )\rangle }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{2}}+{\langle \stackrel{ˆ}{h}(\xi ),u(\xi )\rangle }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{3}}\right\}d\xi \end{array}\) eq 3.1-3
We can get rid of the trade term from [éq 3.1-3] because we assume that \(h(t)\ge 0\text{pp}t\). Using a duality argument, the Cauchy-Schwartz inequality, Lemma 2, and the relationship \(2\text{ab}\le {(\frac{a}{\alpha })}^{2}+{(b\alpha )}^{2}(\alpha >0)\), we get
\(\begin{array}{}\underset{0}{\overset{t}{\int }}{\langle \stackrel{ˆ}{s}(\xi ),u(\xi )\rangle }_{\text{-}1\times \mathrm{1,}\Omega }d\xi \le \frac{1}{2}(\frac{1}{{\alpha }^{2}}\underset{0}{\overset{t}{\int }}{\parallel \stackrel{ˆ}{s}(\xi )\parallel }_{\text{-}\mathrm{1,}\Omega }^{2}d\xi +\frac{{P}^{2}(\Omega )}{{\parallel \lambda \parallel }_{\infty ,\Omega }}{\alpha }^{2}\underset{0}{\overset{t}{\int }}{\parallel \sqrt{\lambda }\nabla u(\xi )\parallel }_{\mathrm{0,}\Omega }^{2}d\xi )\\ \end{array}\) eq 3.1-4
We perform the same work on the loads, thus defining the parameters \(\beta ` and :math:\)gamma` by using the notations of Theorem 3 (for \({C}_{i}\)…), then we insert these inequalities into [éq 3.1-3]
\(\begin{array}{}{\parallel \mid \overline{\rho }{C}_{p}u(t)\parallel }_{\mathrm{0,}\Omega }^{2}+(2-\frac{{P}^{2}(\Omega )}{{\parallel \lambda \parallel }_{\infty ,\Omega }}({\alpha }^{2}+{C}_{2}^{2}{\beta }^{2}+{C}_{3}^{2}{\gamma }^{2}))\underset{0}{\overset{t}{\int }}{\parallel \sqrt{\lambda }\nabla u(\xi )\parallel }_{\mathrm{0,}\Omega }^{2}d\xi \le \\ {\parallel \mid \overline{\rho }{C}_{p}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}+\underset{0}{\overset{t}{\int }}\left\{\frac{{\parallel \stackrel{ˆ}{s}(\xi )\parallel }_{\text{-}1\times \mathrm{1,}\Omega }^{2}}{{\alpha }^{2}}+\frac{{\parallel \stackrel{ˆ}{g}(\xi )\parallel }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{2}}^{2}}{{\beta }^{2}}+\frac{{\parallel \stackrel{ˆ}{h}(\xi )\parallel }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{3}}^{2}}{{\gamma }^{2}}\right\}d\xi \end{array}\) eq 3.1-5
It now remains to look for a triplet of strictly positive reals \((\alpha ,\beta ,\gamma )\), not preferring any particular term, in order to reveal a constant independent of the solution and the configuration in front of the gradient term. One arbitrarily chooses to pose
\(2-\frac{{P}^{2}(\Omega )}{{\parallel \lambda \parallel }_{\infty ,\Omega }}({\alpha }^{2}+{C}_{2}^{2}{\beta }^{2}+{C}_{3}^{2}{\gamma }^{2})=1\) eq 3.1-6
Or, for example,
\(\{\begin{array}{c}{\alpha }^{2}=\frac{{\parallel \lambda \parallel }_{\infty ,\Omega }(\text{mes}({\Gamma }_{1})+1)}{{P}^{2}(\Omega )(\text{mes}(\Gamma )+3)}\\ {\beta }^{2}=\frac{{\parallel \lambda \parallel }_{\infty ,\Omega }(\text{mes}({\Gamma }_{2})+1)}{{C}_{2}^{2}{P}^{2}(\Omega )(\text{mes}(\Gamma )+3)}\\ {\gamma }^{2}=\frac{{\parallel \lambda \parallel }_{\infty ,\Omega }(\text{mes}({\Gamma }_{3})+1)}{{C}_{3}^{2}{P}^{2}(\Omega )(\text{mes}(\Gamma )+3)}\end{array}\) eq 3.1-7
Hence the increase [éq 3.1-1] by taking
\({K}_{1}=\frac{{P}^{2}(\Omega )(\text{mes}(\Gamma )+3)}{{\parallel \lambda \parallel }_{\infty ,\Omega }}\text{max}(\frac{1}{(\text{mes}({\Gamma }_{1})+1)},\frac{{C}_{2}^{2}}{(\text{mes}({\Gamma }_{2})+1)},\frac{{C}_{3}^{2}}{(\text{mes}({\Gamma }_{3})+1)})\) eq 3.1-8
Notes:
3.2. Semi-discretization in time#
We set a time step \(\Deltat\) such that \(\frac{\tau }{\Deltat }\) is an integer \(N\) and such that the time discretization is regular: \({t}_{0}=\mathrm{0,}{t}_{1}=\Deltat ,{t}_{2}=2\Deltat \cdots {t}_{n}=n\Deltat\).
Note:
This assumption of regularity is not really important, it just makes it possible to simplify the writing of the semi-discretized problem. To model any transient at this time \({t}_{n}\) , all you have to do is replace \(\Delta t\) with \(\Delta {t}_{n}={t}_{n\text{+}1}-{t}_{n}\) .
The semi‑time discretization**of [éq 2.1-1] by the :math:`theta`**-method** leads to the following problem:
We’re looking for the next one
\({({u}^{n})}_{0\le n\le N}\in V\) eq 3.2-1
Such as
\(({P}_{1}^{n\text{+}1})\{\begin{array}{c}\rho {C}_{p}\frac{{u}^{n\text{+}1}-{u}^{n}}{\Delta t}-\theta \text{div}(\lambda \nabla {u}^{n\text{+}1})-(1-\theta )\text{div}(\lambda \nabla {u}^{n})=\theta {\stackrel{ˆ}{s}}^{n\text{+}1}+(1-\theta ){\stackrel{ˆ}{s}}^{n}\Omega 0\le n\le N-1\\ {u}^{n\text{+}1}=0{\Gamma }_{1}0\le n\le N-1\\ \lambda \frac{\partial {u}^{n\text{+}1}}{\partial n}={\stackrel{ˆ}{g}}^{n\text{+}1}{\Gamma }_{2}0\le n\le N-1\\ \lambda \frac{\partial {u}^{n\text{+}1}}{\partial n}+{h}^{n\text{+}1}{u}^{n\text{+}1}={\stackrel{ˆ}{h}}^{n\text{+}1}{\Gamma }_{3}0\le n\le N-1\\ {u}^{0}(\text{.})={u}_{0}\Omega \end{array}\) eq 3.2-2
By posing
\({\Xi }^{n}=\Xi (x,n\frac{\tau }{\Deltat })\text{avec}\Xi \in \left\{u,\stackrel{ˆ}{s},\stackrel{ˆ}{h},h,\stackrel{ˆ}{g}\right\}\text{et}0\le n\le N\)
By multiplying [éq 3.2-2] by a \(v\) test function and integrating on \(\Omega\), we find (via Green’s formula) of course the variational formulation [éq 2.2-3] semi-discretized in time
\(({P}_{2}^{n\text{+}1})\{\begin{array}{c}\text{Etant donnés}{u}^{n},{\stackrel{ˆ}{s}}^{n},{\stackrel{ˆ}{s}}^{n\text{+}1},{\stackrel{ˆ}{g}}^{n},{\stackrel{ˆ}{g}}^{n\text{+}1},{\stackrel{ˆ}{h}}^{n},{\stackrel{ˆ}{h}}^{n\text{+}1},{h}^{n},{h}^{n\text{+}1}\\ \text{Calculer}{u}^{n\text{+}1}\in V\text{tel que}\\ {(\rho {C}_{p}{u}^{n\text{+}1},\nu )}_{\mathrm{0,}\Omega }\text{+}\Delta ta(n\Delta t\theta ;{u}_{\theta }^{n\text{+}1},\nu )={(\rho {C}_{p}{u}^{n},\nu )}_{\mathrm{0,}\Omega }\text{+}\Delta t({b}_{\theta }^{n},\nu )(\forall \nu \in V)\end{array}\) eq 3.2-3
with
\(\begin{array}{}{\Xi }_{\theta }^{n\text{+}1}:=\theta {\Xi }^{n\text{+}1}\text{+}(1\text{-}\theta ){\Xi }^{n}\text{où}\Xi \in \left\{u,\text{hu},b,\stackrel{ˆ}{s},\stackrel{ˆ}{g},\stackrel{ˆ}{h}\right\}\\ a(n\Delta t\theta ;{u}_{\theta }^{n\text{+}1},\nu ):=\underset{\Omega }{\int }\lambda \nabla {u}_{\theta }^{n\text{+}1}\text{.}\nabla \nu \text{dx}\text{+}\underset{{\Gamma }_{3}}{\int }{(\text{hu})}_{\theta }^{n\text{+}1}\nu d\sigma \\ ({b}_{\theta }^{n\text{+}1},\nu ):={\langle {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1},\nu \rangle }_{\text{-}1\times \mathrm{1,}\Omega }\text{+}{\langle {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1},{\gamma }_{\mathrm{0,2}}\nu \rangle }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{2}}\text{+}{\langle {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1},{\gamma }_{\mathrm{0,3}}\nu \rangle }_{\text{-}\frac{1}{2}\times \frac{1}{2},{\Gamma }_{3}}\end{array}\) eq 3.2-4
This semi-discretization in time made it possible to transform our parabolic problem into an elliptic problem to which we can apply the standard Lax-Milgram theorem. The hypotheses of this theorem are easily verified thanks to the continuity and ellipticity results of the proof of theorem 3. Hence the existence and the uniqueness of the sought after suite \({({u}^{n})}_{0\le n\le N}\in V\).
Notes:
By asking \(\text{Rf}=0\) we find the semi-discretized variational formulation of the Code_Aster (cf. [R5.02.01 §5.1.3]). The Dirichlet condition (s) (generalized or not) are verified in the W workspace to which the solution must belong. In addition, by fully implicating the \(\theta\) -method (retrograde Euler) we find the formulation of the SYRTHES [bib9] code.
To be able to semi-discretize by the \(\theta\) -method we need to restrict the membership of the new source to \(\stackrel{ˆ}{s}\in {C}^{0}(\mathrm{0,}\tau ;{H}^{\text{-}1}(\Omega ))\) (to be able to take a value in a given moment). On the other hand, initializing the iterative process [éq 3.2-3] necessarily results in \({u}_{0}\in {H}^{1}(O)\) .
To simplify the expressions, we will no longer mention the time dependence of the bilinear form a (t;.,.) * (for the implicitation of the exchange term), it will remain implied by that of the solution.
As for the continuing problem, by not making any concessions on the assumptions of regularity, we have the following « low » increase:
Property 6
Assuming that the assumptions of property 5 are true, that the \(\theta\) - schema is unconditionally stable (\(\mathrm{\theta }\ge \frac{1}{2}\)), that \(\stackrel{ˆ}{s}\in {C}^{0}(\mathrm{0,}\tau ;{H}^{\text{-}1}(\Omega ))\) and \({u}_{0}\in {H}^{1}(O)\), we have the « low » controllability of the semi-discretized problem in time (with \({K}_{1}({\parallel \lambda \parallel }_{\infty ,\Omega },\text{mes}({\Gamma }_{i}),{\gamma }_{\mathrm{0,}i},P(\Omega ))>0\))
\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}{u}^{n\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t{\parallel \sqrt{\lambda }\nabla {u}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\le \frac{1}{2}{\parallel \sqrt{\rho {C}_{p}}{u}^{n\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\frac{4\theta \text{-}3}{2}{\parallel \sqrt{\rho {C}_{p}}{u}^{n}\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\text{-}1\text{+}\frac{\Delta t}{2}{\parallel \sqrt{\lambda }\nabla {u}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\frac{{K}_{1}\Delta t}{2}({\parallel {\stackrel{ˆ}{s}}_{\theta }^{n\text{+}1}\parallel }_{\text{-}\mathrm{1,}\Omega }^{2}\text{+}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{n\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}^{2}\text{+}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{n\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}^{2})\end{array}\) eq 3.2-5
Proof:
This inequality is easily obtained by repeating the steps described in the demonstration of property 5. On the other hand, you have to multiply [éq 3.2-2] by the particular test function
\({u}_{\theta }^{n\text{+}1}:=\theta {u}^{n\text{+}1}+(1-\theta ){u}^{n}\in V\) eq 3.2-6
and rule out the trade term by argument
\(0<\text{min}({h}^{n},{h}^{n\text{+}1}){\parallel {u}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}{\Gamma }_{3}}^{2}\le \underset{{\Gamma }_{3}}{\int }{(\text{hu})}_{\theta }^{n\text{+}1}{u}_{\theta }^{n\text{+}1}\text{dx}\le \text{max}({h}^{n},{h}^{n\text{+}1}){\parallel {u}_{\theta }^{n\text{+}1}\parallel }_{\mathrm{0,}{\Gamma }_{3}}^{2}\) eq 3.2-7
On the other hand, this time it is not only the source term and the loads that require the trick [éq3.1-4], it is also necessary to put it in place on the \((2\theta -1)\underset{\Omega }{\int }\rho {C}_{p}{u}^{n\text{+}1}{u}^{n}\text{dx}\) cross term. Hence a fourth parameter \(\eta\) verifying a system of the type [éq 3.1-6]
\(\begin{array}{}\mid 2-\frac{{P}^{2}(\Omega )}{{\parallel \lambda \parallel }_{\infty ,\Omega }}({\alpha }^{2}+{C}_{2}^{2}{\beta }^{2}+{C}_{3}^{2}{\gamma }^{2})=1\\ \mid 2\theta -{\eta }^{2}\mid 1-2\theta \mid =1\end{array}\) eq 3.2-8
Notes:
If we do not place ourselves in the case of a conditionally stable schema, in addition to the numerical problems that may arise during the effective implementation of the operator, we will not be able to determine the parameters \((\alpha ,\beta ,\gamma ,\eta )\) governing the equation [éq 3.2-8].
By placing ourselves in the particular framework [:ref:`éq 3.1-9 <éq 3.1-9>`] of article [:ref:`bib6 <bib6>`] and by using equivalent standards [:ref:`éq 3.1-10 <éq 3.1-10>`], such as :math:`frac{4theta -3}{2}<frac{1}{2}`, we find the inequality (5) pp428. *
By saying [éq 3.2-5] for the values of \(m\in \left\{\mathrm{0,1}\dots ,n\right\}\) and summing these increases up to \(n\), we get the next « low » increase that takes into account the history of the solutions and the data.
Corollary 7
Under the assumptions of property 6, we have the increase
\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}{u}^{n}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla {u}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}4(1\text{-}\theta )\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\rho {C}_{p}}{u}^{m}\parallel }_{\mathrm{0,}\Omega }^{2}\le (4\theta \text{-}3){\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\text{+}{K}_{1}\Delta t\sum _{m=0}^{n\text{}-\text{}1}({\parallel {\stackrel{ˆ}{s}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\mathrm{1,}\Omega }^{2}\text{+}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}^{2}\text{+}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}^{2})\end{array}\) eq 3.2-9
or more simply
\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}{u}^{n}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla {u}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\le {\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\text{+}{K}_{1}\Delta t\sum _{m=0}^{n\text{-}1}({\parallel {\stackrel{ˆ}{s}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\mathrm{1,}\Omega }^{2}\text{+}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}^{2}\text{+}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}^{2})\end{array}\) eq 3.2-10
Proof:
Since obtaining [éq 3.2-9] has already been explained, it remains to be demonstrated [éq 3.2-10]. This more « gross » inequality simply comes from the fact that
\(\begin{array}{}4(1-\theta )\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\rho {C}_{p}}{u}^{m}\parallel }_{\mathrm{0,}\Omega }^{2}\ge 0\\ (4\theta -3){\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}\le {\parallel \sqrt{\rho {C}_{p}}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}\end{array}\) eq 3.2-11
Notes:
We can obviously make the same remark as [bib6] by noting that the last term of [éq3.2-9] is a Riemann sum that tends to the last term of [éq 3.1-1] when the time step approaches zero. On the other hand, if we introduce the function (with \({\chi }_{\left[n\Delta t,(n\text{+}1)\Delta t\right]}\) the time function characteristic of the interval \(\left[n\Delta t,(n+1)\Delta t\right]\)) \(u(t)={u}_{\theta }^{n\text{+}1}{\chi }_{\left[n\Delta t,(n\text{+}1)\Delta t\right]}(t)\) refines piecewise into [éq 3.1-1], we find exactly [éq3.2-9].
*As with [éq 3.1-1], taking the less restrictive hypotheses (H*4) and (H5), we find a « strong » version of properties 6 and 7. *
3.3. Temporal discretization error#
The previous results on the continuous problem and on its semi-discretized form in time are reused jointly to study the controllability of the temporal discretization error.
\(\begin{array}{}\forall 0\le n\le N{e}^{n}:={u}^{n}-u(n\Deltat )\\ {e}^{0}=0\end{array}\) eq 3.3-1
We start by making this error appear by subtracting from equation [éq 3.2-2] the relationships
\(\begin{array}{}\frac{1}{\Delta t}\underset{n\Delta t}{\overset{(n\text{+}1)\Delta t}{\int }}\frac{\partial u(\xi )}{\partial t}d\xi =\frac{u((n\text{+}1)\Delta t)\text{-}u(n\Delta t)}{\Delta t}\\ \theta \rho {C}_{p}\frac{\partial u((n\text{+}1)\Delta t)}{\partial t}=\theta \text{div}(\lambda \nabla u((n\text{+}1)\Delta t))\text{+}\theta \stackrel{ˆ}{s}((n\text{+}1)\Delta t)\\ (1\text{-}\theta )\rho {C}_{p}\frac{\partial u(n\Delta t)}{\partial t}=(1\text{-}\theta )\text{div}(\lambda \nabla u(n\Delta t))\text{+}(1\text{-}\theta )\stackrel{ˆ}{s}(n\Delta t)\end{array}\) eq 3.3-2
either
\(\rho {C}_{p}\frac{{e}^{n\text{+}1}\text{-}{e}^{n}}{\Delta t}\text{-}\text{div}(\lambda \nabla {e}_{\theta }^{n\text{+}1})=\frac{1}{\Delta t}\underset{n\Delta t}{\overset{(n\text{+}1)\Delta t}{\int }}\frac{\partial u(\xi )}{\partial t}d\xi \text{+}\rho {C}_{p}{(\frac{\partial u}{\partial t})}_{\theta }\) eq 3.3-3
by noting
\(\begin{array}{}{e}_{\theta }^{n\text{+}1}:=\theta {e}^{n\text{+}1}\text{+}(1\text{-}\theta ){e}^{n}\\ {(\frac{\partial u}{\partial t})}_{\theta }:=\theta \frac{\partial u}{\partial t}((n\text{+}1)\Delta t)\text{+}(1\text{-}\theta )\frac{\partial u}{\partial t}(n\Delta t)\end{array}\) eq 3.3-4
From this expression we can describe, via the use of the Taylor formula, the « low » controllability of the time discretization error. But to be able to use the time derivatives of the continuous solution we need a minimum of regularity in \(t\), for example by conceding that
\(u\in {H}^{1}(\mathrm{0,}\tau ;V)\cap {H}^{2}(\mathrm{0,}\tau ;{H}^{\text{-}1}(\Omega ))\) eq 3.3-5
Property 8
Assuming that the solution verifies the additional hypothesis of temporal regularity [éq 3.3-5], we have the « low » controllability of the temporal discretization error
\(\begin{array}{}\forall 0\le n\le N{\parallel \sqrt{\rho {C}_{p}}{e}^{n}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla {e}_{\theta }^{m\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\le \\ \frac{{K}_{1}{(\Delta t)}^{3}{(\rho {C}_{p})}^{2}}{4}\sum _{m=0}^{n\text{-}1}((1\text{-}\theta )\frac{{\partial }^{2}u}{\partial {t}^{2}}(m\Delta t)\text{-}\theta \frac{{\partial }^{2}u}{\partial {t}^{2}}((m\text{+}1)\Delta t))\end{array}\) eq 3.3-6
Proof:
By evaluating [éq 3.3-3] by a Taylor formula in 2nd order, we involve the second time derivative of the solution and we show that the error sequence \({({e}^{n})}_{0\le n\le N}\in V\) verifies a problem similar to [éq 3.2-2] (assuming that the temporal discretization of the boundary conditions is exact)
\(({P}_{3}^{n\text{+}1})\{\begin{array}{c}\begin{array}{}\rho {C}_{p}\frac{{e}^{n\text{+}1}\text{-}{e}^{n}}{\Delta t}\text{-}\text{div}(\lambda \nabla {e}_{\theta }^{n\text{+}1})=\\ \frac{\rho {C}_{p}\Delta t}{2}((1\text{-}\theta )\frac{{\partial }^{2}u}{\partial {t}^{2}}(n\Delta t)\text{-}\theta \frac{{\partial }^{2}u}{\partial {t}^{2}}((n\text{+}1)\Delta t))\Omega 0\le n\le N\text{-}1\end{array}\\ {e}^{n\text{+}1}=0{\Gamma }_{1}0\le n\le N\text{-}1\\ \lambda \frac{\partial {e}^{n\text{+}1}}{\partial n}=0{\Gamma }_{2}0\le n\le N\text{-}1\\ \lambda \frac{\partial {e}^{n\text{+}1}}{\partial n}\text{+}{h}^{n\text{+}1}{e}^{n\text{+}1}=0{\Gamma }_{3}0\le n\le N\text{-}1\\ {e}^{0}(\text{.})=0\Omega \end{array}\) eq 3.3-7
We can then apply to it the second result of corollary 7 from where [éq 3.3-6] (we could, of course, have just as easily applied the raw result of this corollary or that of property 6 from which it derives).
Notes:
By placing ourselves in the particular framework [éq 3.1-9] of article [bib6] with an implicit diagram \((\theta =1)\) and by using the equivalent standards [éq 3.1-10] we find the inequality (8) pp429. All you have to do is make \(\Deltat \to 0\) tense and approximate the integral by the Riemann sum of the second member of [éq 3.3-6].
The existence and uniqueness of the sequence \(({e}^{n})\) follows of course from that of \(({u}^{n})\) but it can also be redemonstrated by applying the Lax-Milgram theorem to the weak formulation arising from [éq3.3-7].
3.4. Total discretization in time and space#
It is assumed that domain \(\Omega\) is polyhedral or not and that it is spatially discretized by a regular family \({({T}_{h})}_{h}\) of triangulations. Because of this regularity, the finite element method applied to \(({P}_{2}^{n\text{+}1})\) converges when the largest diameter of the \(K\) elements in \({({T}_{h})}_{h}\) approaches zero.
\(h:=\underset{K\in {T}_{h}}{\text{max}}{h}_{K}\to 0\) eq 3.4-1
Notes:
The finite elements \((K,{P}_{K},{\Sigma }_{K})\) are affine equivalent to the same reference elements, they verify compatibility relationships on their common borders and geometric constraints [éq 3.4-1] and [éq 3.4-2].
Remember that the diameter of K is the real \({h}_{K}:=\underset{(x,y)\in {K}^{2}}{\text{max}\mid x-y\mid }\) .
By noting \({\rho }_{K}\) the roundness (remember that the roundness of \(K\) is the real \({\rho }_{K}:=\text{max}\left\{\text{diamètre des sphères}\subset K\right\}\)) associated with \(K\), the finite elements of \({({T}_{h})}_{h}\) also satisfy the constraint
\(\exists \sigma >0/\frac{{h}_{K}}{{\rho }_{K}}\le \sigma\) eq 3.4-2
In the usual triplet \((K,{P}_{K},{\Sigma }_{K})\), we define the polynomial space as being that of polynomials with a total degree less than or equal to \(k\) out of \(K\).
\({P}_{K}:={P}_{k}(K)\) eq 3.4-3
and the associated approximation space (in the « weak » sense)
\({V}_{h}:=\left\{{\nu }_{h}\in V/\forall K\in {T}_{h}{\nu }_{{h}_{\mid K}}\in {P}_{k}(K)\right\}\subset V\) eq 3.4-4
To conclude, note \({\Pi }_{h}\), the projection operator that associates the continuous solution with its \({V}_{h}\) —interpolated
\(\begin{array}{}{\mathrm{\Pi }}_{h}:V\to {V}_{h}\\ \nu \to {\nu }_{h}\end{array}\) eq 3.4-5
Note:
With a regular family of triangulations, this interpolation operator is continuous and can be written \({\mathrm{\Pi }}_{h}\nu =\sum _{i}\nu ({x}_{i}){N}_{i}\) notating \({x}_{i}\) the vertices of the mesh and \({N}_{i}\) their associated shape function.
It will be of particular importance when describing the increase that will unearth the error indicator.
Notes:
In the definition [:ref:`éq 3.4-4 <éq 3.4-4>`] of :math:`{V}_{h}`, it is the compatibility relationships intrinsic to the family of elements that assures us* \(\forall h,K{\nu }_{{h}_{\mid K}}\in {P}_{k}(K)\subset {H}^{1}(K)\Rightarrow {\nu }_{h}\in {H}^{1}(\Omega :=\cup \stackrel{ˉ}{K})\) eq 3.4-6 In the literature, the more regular definition is often preferred :math:`{V}_{h}^{text{*}}:={V}_{h}cap {C}^{0}(Omega )`**eq 3.4-7** |
By using the semi-discretized form \(({P}_{2}^{n\text{+}1})\) with test functions in \({V}_{h}\) we obtain the following problem that is completely discretized in time and space (for a fixed \(h\)):
We’re looking for the next one
\({({u}_{h}^{n})}_{0\le n\le N}\in {V}_{h}\) eq 3.4-8
initialized by
\({u}_{h}^{0}:={\mathrm{\Pi }}_{h}{u}_{0}\) eq 3.4-9
verifying the following problem
\(({P}_{2}^{h,n\text{+}1})\{\begin{array}{c}\text{Etant donnés}{u}_{h}^{n},{\stackrel{ˆ}{s}}^{n},{\stackrel{ˆ}{s}}^{n\text{+}1},{\stackrel{ˆ}{g}}^{n},{\stackrel{ˆ}{g}}^{n\text{+}1},{\stackrel{ˆ}{h}}^{n},{\stackrel{ˆ}{h}}^{n\text{+}1},{h}^{n},{h}^{n\text{+}1}\\ \text{Calculer}{u}_{h}^{n\text{+}1}\in {V}_{h}\text{tel que}\\ {(\rho {C}_{p}{u}_{h}^{n\text{+}1},{\nu }_{h})}_{\mathrm{0,}\Omega }\text{+}\Delta ta({u}_{\theta ,h}^{n\text{+}1},{\nu }_{h})={(\rho {C}_{p}{u}_{h}^{n},{\nu }_{h})}_{\mathrm{0,}\Omega }\text{+}\Delta t({b}_{\theta }^{n\text{+}1},{\nu }_{h})(\forall {\nu }_{h}\in {V}_{h})\end{array}\) eq 3.4-10
Just as it was assumed in the previous paragraph that the discretization of loading times was correct
: label: EQ-None
{e} _ {mathrm {khi}}} ^ {n}} := {xi} ^ {n} -Xi (nnu t) =0text {with}Xiininleft{stackrel {}} {s} {s},stackrel {} {s} {s},stackrel {} {h}, h,stackrel {} {g}right}left{stackrel {} {left{left{stackrel {} {s} {s},stackrel {} {s},stackrel {} {s},stackrel {} {s} {s},stackrel {} {s}} 0le nle N
, we also assume here that their spatial discretization is
: label: EQ-None
forall h {Xi} _ {h} ^ {n} := {mathrm {Pi}}} _ {h} {Xi} ^ {n} = {Xi} ^ {n}text {with}Xi^ {n} {with}text {with}Xiinininininleftleft{left{stackrel {} {s}} {s},stackrel {} {h}, h,stackrel {} {g} {g}}right}text {and} 0le nle N
In Code_Aster, these hypotheses may not be verified and we will see that they impact the quality of the residual indicator and its equivalence relationships with the exact error (cf. [§4.3]). In practice, even if you have to deal with this approximation, it is not really a problem as long as the loads are « not too hectic » in terms of time and space.
By applying the standard Lax-Milgram theorem following the framework developed in the demonstration of theorem 3, we show the existence and the uniqueness of the sequence \({({u}_{h}^{n})}_{n}\) in the closed sev (it is therefore a Hilbert, an indispensable prerequisite for the use of the famous theorem) \({V}_{h}\) of Hilbert \(V\). Moreover, by applying the second result of corollary 7 (one could, of course, have just as easily applied the raw result of this corollary or that of property 6 from which it derives), the « weak » controllability of the totally discretized problem takes the following form:
Property 9
Based on the triangulation defined above and assuming that the hypotheses (\({H}_{6}\)) and (\({H}_{7}\)) are true, we have the increase
\(\begin{array}{}{\parallel \sqrt{\rho {C}_{p}}{u}_{h}^{n}\parallel }_{\mathrm{0,}\Omega }^{2}\text{+}\Delta t\sum _{m=0}^{n\text{-}1}{\parallel \sqrt{\lambda }\nabla {u}_{\theta ,h}^{m\text{+}1}\parallel }_{\mathrm{0,}\Omega }^{2}\le {\parallel \sqrt{\rho {C}_{p}}{\Pi }_{h}{u}_{0}\parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall 0\le n\le N\text{+}{K}_{1}\Delta t\sum _{m=0}^{n\text{-}1}({\parallel {\stackrel{ˆ}{s}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\mathrm{1,}\Omega }^{2}\text{+}{\parallel {\stackrel{ˆ}{g}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}^{2}\text{+}{\parallel {\stackrel{ˆ}{h}}_{\theta }^{m\text{+}1}\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}^{2})\end{array}\) eq 3.4-11
noting \({u}_{\theta ,h}^{m\text{+}1}:=\theta {u}_{h}^{m\text{+}1}+(1-\theta ){u}_{h}^{m}\).
Notes:
By placing ourselves in the particular framework [éq 3.1-9] of article [bib6] with an implicit diagram \((\theta =1)\) and by using the equivalent standards [éq 3.1-10] we find the inequality (14) pp430.
Taking the less restrictive hypotheses ( \({H}_{4}\) ) and ( ) and ( \({H}_{5}\) ), we find a « strong » version of this increase involving the standard \({H}^{1}\) of the result field.
Now that we have identified the functional framework ensuring the existence and uniqueness of the discrete solution suite and studying the evolution of the controllability of the problem during discretizations, we will pool these somewhat « ethereal » results to identify the increase where the indicator will occur.