2. The boundary problem#
2.1. background#
We consider a immobile body occupying a connected bounded open \(O\) of \({R}^{q}\) (\(q\) =2 or 3) of border \(\partial \Omega :=\Gamma :=\underset{i=1}{\overset{3}{\cup }}{\Gamma }_{i}\) regular characterized by its heat volume at constant pressure \(\rho {C}_{p}(x)\) (the vector variable \(x\) here symbolizes the couple \((\text{x,y})\) (resp. \((x,y,z)\)) for \(q\) =2 (resp. \(q\) =3)) )) * and its isotropic thermal conductivity coefficient \(\lambda (x)\).
Note:
A possible displacement of the structure will therefore not be taken into account (see THER_NON_LINE_MO [R5.02.04]).
These material data are assumed to be independent of time (modeling THER of Code_Aster) and constants per element (discretization \({P}_{0}\)).
Note:
With modelling THER_FOces features may depend on the weather. From the first versions of the code and before the implementation THER_NON_LINE, it allowed to simulate « pseudo- » non-linearities. Given its rather marginal use, we will not be interested in this modeling as a step.
We are interested in temperature changes at any point \(x\) of the open and at any time \(t\in [\mathrm{0,}\tau [(\tau >0)\), when the body is subject to limit conditions and loads that are independent of temperature but may depend on time. These are volume source \(s(x\text{,t})\), boundary conditions such as imposed temperature \(f(x\text{,t})\) (on the external surface portion \({\Gamma }_{1}\)), imposed normal flow \(g(x\text{,t})\) (on \({\Gamma }_{2}\)) and convective exchange \(h(x\text{,t})\) and \({T}_{\text{ext}}(x\text{,t})\) (on \({\Gamma }_{3}\)).
We thus place ourselves within the framework of application of the operator THER_LINEAIRE [R5.02.01] of the Code_Aster by retaining only the conductive aspects of this linear thermal problem.
Note:
Nonlinearities pose serious theoretical problems [bib2] to demonstrate the existence, uniqueness, and stability of the possible solution. Some are still completely open… But in practice, this does not at all prevent « stretching » the scope of use of the error estimator that will be rigorously unearthed for linear thermal, to non-linear thermal (operator THER_NON_LINE [R5.02.02]).
This mixed boundary problem (of the Cauchy-Dirichlet-Neumann-Robin type (also called Fourier condition) is inhomogeneous, linear and with variable coefficients) is formulated
\(({P}_{0})\{\begin{array}{c}\rho {C}_{p}\frac{\partial T}{\partial t}-\text{div}(\lambda \nabla T)=s\Omega \times ]\mathrm{0,}\tau [\\ T=f{\Gamma }_{1}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial T}{\partial n}=g{\Gamma }_{2}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial T}{\partial n}+\text{hT}={\text{hT}}_{\text{ext}}{\Gamma }_{3}\times ]\mathrm{0,}\tau [\\ T(x\mathrm{,0})={T}_{0}(x)\Omega \end{array}\) eq 2.1-1
Notes:
\(\text{Avec}{\Gamma }_{3}={\Gamma }_{\text{12}}\cup {\Gamma }_{\text{21}},{T}_{i}={T}_{\mid {\Gamma }_{\text{i}\text{j}}}\text{on a}\{\begin{array}{c}\lambda \frac{\partial {T}_{1}}{\partial n}+{\text{hT}}_{1}={\text{hT}}_{2}{\Gamma }_{\text{12}}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial {T}_{2}}{\partial n}+{\text{hT}}_{2}={\text{hT}}_{1}{\Gamma }_{\text{21}}\times ]\mathrm{0,}\tau [\end{array}\) eq 2.1-2 To avoid burdening the problem and insofar as this option is similar to Robin’s condition with the external environment, we will not mention it specifically in the calculations that follow. * The Dirichlet condition can be generalized in the form of linear relationships between degrees of freedom (keywords LIAISON_* ) to simulate, in particular, geometric symmetries of the structure. \(\begin{array}{}\text{Avec}{\Gamma }_{1}={\Gamma }_{\text{12}}\cup {\Gamma }_{\text{21}},{T}_{i}={T}_{\mid {\Gamma }_{\text{ij}}}\text{on a}(\text{LIAISON\_GROUP})\\ \sum _{i}{\beta }_{\mathrm{1i}}{T}_{1}^{i}(x,t)+\sum _{j}{\beta }_{\mathrm{2j}}{T}_{2}^{j}(x,t)=\gamma (x,t)\text{sur}{\Gamma }_{1}\times ]\mathrm{0,}\tau [:ref:\)\ text{ou plus simplement}sum _{i}{beta }_{i}{T}_{i}(x,t)=gamma (x,t)text{sur}{Gamma }_{1}times <\ text{ou plus simplement}sum _{i}{beta }_{i}{T}_{i}(x,t)=gamma (x,t)text{sur}{Gamma }_{1}times >`]mathrm {0,}tau [(text {LIAISON_ DDL})end {array} *eq 2.1-3* *Similarly, the features* *LIAISON_UNIF* *and* * *LIAISON_CHAMNO* *allow you to impose the same (unknown) temperature on a set of nodes. They constitute an overlay of the previous conditions by imposing particular couples* :math:`(beta ,gamma ) . In order not to make the problem more complicated and insofar as these options are only specific cases of the generic Dirichlet condition, we will not mention them specifically in the calculations that follow. * When the material is anisotropic the conductivity is modelled by a diagonal matrix expressed in the orthotropy coordinate system of the material. This does not fundamentally change the following calculations, which only take into account the isotropic case. You just have to be careful not to switch, under Neumann and Robin limit conditions, the dot product with the normal and the multiplication by conductivity. * For a transient calculation , the initial temperature can be chosen in three different ways: by performing a stationary calculation on the first instant, by setting it to a uniform or any value created by a CREA_CHAMP * ** and by restarting from a previous transient calculation. This choice of the Cauchy condition has no impact on the theoretical study that will follow. * * We will not deal with the case where (almost) all loads are multiplied by the same time-dependent function (option FONC_MULT * ), this feature, which is well suited for certain mechanical problems, is not recommended in thermal applications, as it may conflict with the time dependence of loads and, on the other hand, it applies selectively to each of them. Moreover, it was not included in THER_NON_LINE * ) . |
It is shown that the most general and most convenient functional framework for « handling » this parabolic problem is as follows.
For geometry:
\(O\) |
open bounded locally on only one side of its border, |
(H1) |
\(\Gamma\) |
\(q-1\), Lipschitzian, or \({C}^{1}\) dimensional variety per piece |
(H2) |
For data:
: label: EQ-None
begin {array} {} sin {L} ^ {2} (mathrm {0,}tau; {H} ^ {text {-} 1} (Omega)) {T} _ {0}in {L} ^ {2}in {L} ^ {2} (omega)\ fin {L} ^ {2} (mathrm {0,}) {0,}tau; {H} ^ {2} (mathrm {0,}tau; {H} ^ {2} (mathrm {0,}}tau; {H} ^ {2} (mathrm {0,}}tau; {H} ^ {2})} ({Gamma} _ {1}))), gin {L} ^ {2} (mathrm {0,}tau; {H} ^ {text {-}} ({Gamma} _ {2})), {Gamma} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})), {T} _ {2})text {-}} ({Gamma} _ {3}))\rho, {C} _ {p},lambdain {L} ^ {infty} (Omega) hin {L} _ {3}) hin {L} ^ {3}) hin {L} _ {3})in {L} (in {L}} (Omega) hin {L} _ {3}) hin {L} _ {3})in {L} _ {3})in {L} _ {3})in {L} _ {3})in {L} _ {3})in {L} _ {3})in {L} _ {3})in {L} _ {3})}
That allows us to get a solution in the following intersection
\(T\in {L}^{2}(\mathrm{0,}\tau ;{H}^{1}(\Omega ))\cap {C}^{0}(\mathrm{0,}\tau ;{L}^{2}(\Omega ))\) eq 2.1-4
Note:
Be \((X,{\parallel \parallel }_{X})\) a Banach, we note \({L}^{p}(\mathrm{0,}\tau ;X)\) the space of functions \(t\to v(t)\) highly measurable for measuring \(\text{dt}\) such as \({\parallel \nu \parallel }_{\mathrm{0,}\tau ;p,X}={({\int }_{0}^{\tau }{\parallel \nu (t)\parallel }_{X}^{p}\text{dt})}^{\frac{1}{p}}\text{<+}\infty\) . It’s a Banach, so a Hilbert space for the associated standard.
The introduction of these particular « space-time » Hilbert spaces comes from the need to separate variables \(x\) and \(t\). Any function \(u:(x,t)\in {Q}_{\mathrm{\tau }}:=\Omega \times ]\mathrm{0,}\tau [\to u(x,t)\in \Re\) can in fact be identified (using the Fubini theorem) with another function \(\tilde{u}:t\in ]\mathrm{0,}\tau [\to \left\{\tilde{u}(t):x\in \Omega \to \tilde{u}(t)(x)=u(x,t)\right\}\). Since transformation \(u\to \tilde{u}\) is an isomorphism, we will simplify the expressions later by notating u what should have been meant \(\tilde{u}\).
Notes:
Given the formulation [éq 2.1-1], we will therefore focus on a solution belonging to the following functional space:
Note:
This space also includes any « generalized » Dirichlet conditions such as linear relationships between ddls.
\(T\in W:=\left\{u\in {H}^{1}(\Omega )/{\gamma }_{\mathrm{0,1}}\text{u:}={u}_{\mid {\Gamma }_{1}}=f\right\}\) eq 2-1-5
In addition, thanks to the geometric hypotheses (H1) and (H2), there is a bearing operator (composed of the usual bearing operator and the operator for extending by zero outside \({\Gamma }_{1}\)) \(R:{H}^{\frac{1}{2}}({\Gamma }_{1})\to {H}^{1}(\Omega )\) linear, continuous and surjective such as:
\({\gamma }_{\mathrm{0,1}}\text{Rf}=f\forall f\in {H}^{\frac{1}{2}}({\Gamma }_{1})\) eq 2-1-6
We will therefore be able to make the initial problem homogeneous in Dirichlet by only focusing on the solution
\(u\in V:=\left\{u\in {H}^{1}(\Omega )/{\gamma }_{\mathrm{0,1}}u:={u}_{\mid {\Gamma }_{1}}=0\right\}\) eq 2-1-7
resulting from decomposition
\(T:=u+\text{Rf}\) eq 2-1-8
Note:
Be \((X,{\parallel \parallel }_{X})\) a Banach, we note \({L}^{p}(\mathrm{0,}\tau ;X)\) the space of functions \(t\to \nu (t)\) highly measurable for measuring \(\text{dt}\) such as \({\parallel \nu \parallel }_{\mathrm{0,}\tau ;p,X}={({\int }_{0}^{\tau }{\parallel \nu (t)\parallel }_{X}^{p}\text{dt})}^{\frac{1}{p}}\text{<+}\infty\) . It’s a Banach, so a Hilbert space for the associated standard.
This variable change produces the simplified problem in \(u\)
\(({P}_{1})\{\begin{array}{c}\rho {C}_{p}\frac{\partial u}{\partial t}-\text{div}(\lambda \nabla u)=\stackrel{ˆ}{s}\Omega \times ]\mathrm{0,}\tau [\\ u=0{\Gamma }_{1}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial u}{\partial n}=\stackrel{ˆ}{g}{\Gamma }_{2}\times ]\mathrm{0,}\tau [\\ \lambda \frac{\partial u}{\partial n}+\text{hu}=\stackrel{ˆ}{h}{\Gamma }_{3}\times ]\mathrm{0,}\tau [\\ u(0)={u}_{0}\Omega \end{array}\) eq 2-1-9
with the new second member
\(\stackrel{ˆ}{s}:=s-\rho {C}_{p}\frac{\partial \text{Rf}}{\partial t}+\text{div}(\lambda \nabla \text{Rf})\in {L}^{2}(\mathrm{0,}\tau ;{H}^{\text{-}1}(\Omega ))\), eq 2-1-10
The new loads
\(\stackrel{ˆ}{g}:=g-\lambda \frac{\partial \text{Rf}}{\partial n}\in {L}^{2}(\mathrm{0,}\tau ,{H}^{\text{-}}({\Gamma }_{2}))\) eq 2-1-11
\(\stackrel{ˆ}{h}:=h({T}_{\text{ext}}-\text{Rf})-\lambda \frac{\partial \text{Rf}}{\partial n}\in {L}^{2}(\mathrm{0,}\tau ,{H}^{\text{-}}({\Gamma }_{3}))\) eq 2-1-12
and the new initial condition
\({u}_{0}(\text{.}):={T}_{0}(\text{.})-\text{Rf}(\text{.}\mathrm{,0})\in {L}^{2}(O)\) eq 2-1-13
Note:
This **theoretical approach, which may seem a bit « ethereal », has a very concrete foothold in the numerical techniques used to solve this type of problem. It corresponds to a* substitution (this technique is not used in Code_Aster, we prefer the double dualization technique via Lagrange ddls [R3.03.01]) of the Dirichlet boundary conditions . By renumbering the unknowns so that these conditions appear last, the comparison can be schematized in the following matrix form
\(\left[\begin{array}{cc}A& 0\\ 0& \text{Id}\end{array}\right]\left[\begin{array}{c}T\\ {T}_{{\Gamma }_{1}}:={\text{Rf}}_{{\Gamma }_{1}}\end{array}\right]=\left[\begin{array}{c}\stackrel{ˆ}{s}:=s-\sum _{j>J}{a}_{\text{ji}}{f}_{j}\\ f\end{array}\right]\)
Assumptions of regularity on the border also ensure the following good properties for workspaces. We will then be able to place ourselves in the usual abstract variational framework.
Lemma 1
Under the hypotheses (H1) and (H2) the workspaces \(W\) and \(V\) are Hilberts equipped with the norm induced by \({H}^{1}(O)\).
Proof:
The result comes simply from the fact that the trace \({\gamma }_{\mathrm{0,1}}:{H}^{1}(\Omega )\to {L}^{2}({\Gamma }_{1})\) application is composed of the usual linear, continuous and surjective trace application \({\gamma }_{0}:{H}^{1}(\Omega )\to {H}^{\frac{1}{2}}(\Gamma )\subset {L}^{2}(\Gamma )\) (taking into account the hypotheses adopted) and the \({\Gamma }_{1}\) restriction operator which is also linear, continuous and surjective. From their definition, we deduce that \(W\) and \(V\) are closed sevs of \({H}^{1}(\Omega )\). They are therefore Hilberts equipped with the \({\parallel \parallel }_{\mathrm{1,}\Omega }\) standard.
Lemma 2
Under the hypotheses (\({H}_{1}\)) and (\({H}_{2}\)), the norm and the semi-norm induced by \({H}^{1}(O)\) are equivalent on the functional space \(V\). Note \(P(O)>0\) the Poincaré constant relaying this equivalence.
\(\forall \nu \in V{\parallel \nu \parallel }_{\mathrm{1,}\Omega }\le P(\Omega ){\mid \nu \mid }_{\mathrm{1,}\Omega }\)
Note:
We will subsequently note \({\parallel u\parallel }_{\infty ,\Omega }:=\underset{\text{pp}\text{.}t\in \Omega }{\text{supess}\mid u(t)\mid }\) and \(\forall (u,\nu )\in {({H}^{m}(\Omega ))}^{2}{(u,\nu )}_{m,\Omega }:=\sum _{\mid \alpha \mid \le m}{({\partial }^{\alpha }u,{\partial }^{\alpha }\nu )}_{{L}^{2}(\Omega )},{\parallel u\parallel }_{m,\Omega }^{2}:=\sum _{\mid \alpha \mid \le m}{\parallel {\partial }^{\alpha }u\parallel }_{{L}^{2}(\Omega )}^{2}\text{et}{\mid u\mid }_{m,\Omega }^{2}:=\sum _{\mid \alpha \mid =m}{\parallel {\partial }^{\alpha }u\parallel }_{{L}^{2}(\Omega )}^{2}\).
Proof:
This result is a corollary of the Poincaré inequality verified by the open words of « Nikodym », of which \(\Omega\) is a part, taking into account the hypotheses adopted. However, there are two scenarios:
or the problem is really mixed and has boundary conditions other than those of Dirichlet, \(\text{mes}(\Gamma -{\Gamma }_{1})\ne 0\) (see the demonstration [bib1] § III .7.2 pp922-925),
or we only take into account conditions such as imposed temperature, \(\text{mes}(\Gamma -{\Gamma }_{1})=0\), \(V={H}_{0}^{1}(\Omega )\) and we find the standard result of equivalence of the norm and the semi-norm on this space (see for example the demonstration [bib3] pp18-19).
The compilation of the previous results makes it possible to identify the Abstract Variational Framework (CVA) on which the weak formulation will be based:
\({H}_{0}^{1}(\Omega )\subset V\subset {H}^{1}(\Omega )\),
\(V\subset H:={L}^{2}(\Omega )=H\text{'}\subset V\text{'}\subset {H}^{\text{-}1}(\Omega )\) by identifying \(H\) and its dual,
we have a continuous linear canonical injection of \(V\) into \(H\),
\(V\) is dense in \(H\) and the injection is compact (it inherits the properties of \({H}^{1}(\Omega )\) compared to \(H\)),
\(V\) is equipped with the semi-norm induced by \({H}^{1}(\Omega )\) and \(H\) with its usual norm.
Note:
Based on a formulation of the Rellich compactness theorem adapted to Sobolev spaces on an open space (for example, theorem 1.5.2 [bib3] pp29-30).
2.2. From strong to weak formulations#
By multiplying the main equation of the boundary problem [éq 2.1-1] by a \(\nu \in V\) test function and using the Green and Reynolds theorems (to switch the integral to space and the derivation to time, with \(\Omega\) fixed and material characteristics independent of time), we obtain:
\(\frac{d}{\text{dt}}\underset{\Omega }{\int }\rho {C}_{p}u(t)\nu \text{dx}+\underset{\Omega }{\int }\lambda \nabla u(t)\text{.}\nabla \nu \text{dx}=\underset{\Omega }{\int }\stackrel{ˆ}{s}(t)\nu \text{dx}+\underset{\Gamma }{\int }\mathrm{\lambda }\frac{\partial u(t)}{\partial n}\nu d\sigma\) eq 2.2-1
By introducing the boundary conditions in [éq 2.2-1], the following weak formulation (in the sense of of**of the distributions (in this general framework, the time derivative is therefore to be taken in the weak sense) temporal of \(D\text{'}(]\mathrm{0,}\tau [)\)) is obtained:
We are looking for the solution
\(u\in {L}^{2}(\mathrm{0,}\tau ;V)\cap {C}^{0}(\mathrm{0,}\tau ;H)\) eq 2.2-2
verifying the problem
\(({P}_{2})\left\{\begin{array}{}\mathrm{trouver}u:t\in \text{]}\mathrm{0,}\tau \text{[}\to u(t)\in V\mathrm{tel}\mathrm{que}\\ \forall \nu \in V\frac{d}{\mathrm{dt}}{(\rho {C}_{p}u(t),\nu )}_{\mathrm{0,}\Omega }+a(t;u(t),\nu )=(b(t),\nu )\\ u(0)={u}^{0}\end{array}\right\}\)
Eq 2.2-3
with
\(\begin{array}{}a(t;u(t),\nu ):=\underset{\Omega }{\int }\lambda \nabla u(t)\text{.}\nabla \nu \text{dx}+\underset{{\Gamma }_{3}}{\int }h(t){\gamma }_{\mathrm{0,3}}u(t){\gamma }_{\mathrm{0,3}}\nu d\sigma \\ (b(t),\nu ):={\langle \stackrel{ˆ}{s}(t),\nu \rangle }_{-1\times \mathrm{1,}\Omega }+{\langle \stackrel{ˆ}{g}(t),{\gamma }_{\mathrm{0,2}}v\rangle }_{-\frac{1}{2}\times \frac{1}{2},{\Gamma }_{2}}+{\langle \stackrel{ˆ}{h}(t),{\gamma }_{\mathrm{0,3}}\nu \rangle }_{-\frac{1}{2}\times \frac{1}{2},{\Gamma }_{3}}\end{array}\) eq 2.2-4
by noting \({\langle ,\rangle }_{p\times q,\mathrm{\Theta }}\) the duality hook between spaces \({H}^{p}(\Theta )\) and \({H}^{q}(\mathrm{\Theta })\).
Notes:
\(\stackrel{ˆ}{s}\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}(\Omega )),\stackrel{ˆ}{g}\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}({\Gamma }_{2}))\text{et}\stackrel{ˆ}{h}\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}({\Gamma }_{3}))\) eq 2.2-5 According to [éq 2-1-10] [éq 2-1-12] this restriction can be reflected on initial uploads in the form \(f\in {L}^{2}(\mathrm{0,}\tau ;{H}^{\frac{3}{2}}({\mathrm{\Gamma }}_{1})),s\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}(\Omega )),g\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}({\mathrm{\Gamma }}_{2}))\text{et}{T}_{\text{ext}}\in {L}^{2}(\mathrm{0,}\tau ;{L}^{2}({\mathrm{\Gamma }}_{3}))\) eq 2.2-6 * The formulation \(({P}_{2})\) makes sense, because we show that \(t\to a(t;u(t),\nu )\in {L}^{2}(]\mathrm{0,}\tau [)\subset D\text{'}(]\mathrm{0,}\tau [)\) \(t\to \rho {C}_{p}u(t)\in {L}^{2}(\mathrm{0,}\tau ;V)\text{et}\nu \in V\Rightarrow t\to {(\rho {C}_{p}u(t),\nu )}_{\mathrm{0,}\Omega }\in {L}^{2}(]\mathrm{0,}\tau [)\subset D\text{'}(]\mathrm{0,}\tau [)\) \(\begin{array}{}t\to \stackrel{ˆ}{s}(t)\in {L}^{2}(\mathrm{0,}\tau ;{H}^{\text{-}1}(\Omega ))\text{et}\nu \in {H}^{1}(\Omega )\subset {H}^{\text{-}1}(\Omega )\\ \Rightarrow t\to {\langle \stackrel{ˆ}{s}(t),\nu \rangle }_{-1\times \mathrm{1,}\Omega }\in {L}^{2}(]\mathrm{0,}\tau [)\subset D\text{'}(]\mathrm{0,}\tau [)\end{array}\) \(\begin{array}{}t\to \stackrel{ˆ}{g}(t)\in {L}^{2}(\mathrm{0,}\tau ;{H}^{\text{-}\frac{1}{2}}({\Gamma }_{2}))\text{et}{\gamma }_{\mathrm{0,2}}\nu \in {H}^{\frac{1}{2}}({\Gamma }_{2})\subset {H}^{\text{-}\frac{1}{2}}({\Gamma }_{2})\\ \Rightarrow t\to {\langle \stackrel{ˆ}{g}(t),{\gamma }_{\mathrm{0,2}}\nu \rangle }_{-\frac{1}{2}\times \frac{1}{2},{\Gamma }_{2}}\in {L}^{2}(]\mathrm{0,}\tau [)\subset D\text{'}(]\mathrm{0,}\tau [)\end{array}\) and we obviously find the same thing for the trade term on \({\Gamma }_{3}\) . * In surface integrals we will henceforth note \(u(t)\) and \(v\) what should be noted (from a point of view) \({\mathrm{\gamma }}_{\mathrm{0,}i}u(t)\text{et}{\mathrm{\gamma }}_{\mathrm{0,}i}v\) . * The belonging of the solution to \({L}^{2}(\mathrm{0,}\tau ;V)\) derives from assumptions about the data and from the properties of the differential and trace operators. The fact that it must also belong to \({C}^{0}(\mathrm{0,}\tau ;H)\) comes just from the necessary justification of the Cauchy condition. |
We can then focus on the existence and the uniqueness of the solution of the initial problem \(({P}_{0})\) by showing its equivalence with \(({P}_{2})\) and by applying to the latter a parabolic variant of the Lax-Milgram theorem.
Theorem 3
In the abstract variational framework (CVA) defined above and assuming that the hypotheses (H1), (H2) and (H3) are true, then the problem \(({P}_{2})\) admits one solution and only one \(u\in {L}^{2}(\mathrm{0,}\tau ;V)\cap {C}^{0}(\mathrm{0,}\tau ;H)\).
Proof:
This result comes from the theorems 1 & 2 of the « Dautray-Lions » (cf. [bib3], § XVIII pp615-627). To use them, however, you must check
The measurability of the bilinear form \(\forall (u(t),\nu )\in {V}^{2}t\to a(t;u(t),v)\text{sur}]\mathrm{0,}\tau [\)
Its continuity on \(V\times V\)
\(\begin{array}{}\text{pp}t\in ]\mathrm{0,}\tau [\mid a(t;u(t),\nu )\mid \le {\parallel \lambda \parallel }_{\infty ,\Omega }{\mid u(t)\mid }_{\mathrm{1,}\Omega }{\mid \nu \mid }_{\mathrm{1,}\Omega }+{\parallel h(t)\parallel }_{\infty ,{\Gamma }_{3}}{\parallel u(t)\parallel }_{\frac{1}{2},{\Gamma }_{3}}{\parallel \nu \parallel }_{\frac{1}{2},{\Gamma }_{3}}\\ \forall (u(t),\nu )\in {V}^{2}\le \text{max}({\parallel \lambda \parallel }_{\infty ,\Omega },{\parallel h(t)\parallel }_{\infty ,{\Gamma }_{3}}{C}_{3}^{2}{P}^{2}(\Omega )){\mid u(t)\mid }_{\mathrm{1,}\Omega }{\mid \nu \mid }_{\mathrm{1,}\Omega }\end{array}\)
with \({C}_{3}\) the continuity constant of the trace operator on \({\Gamma }_{3}\) and \(P(\Omega )\) the Poincaré constant.
Its \(V\) -ellipticity compared to \(H\)
\(\begin{array}{}\text{pp}t\in ]\mathrm{0,}\tau [a(t;\nu ,\nu )+\frac{\beta }{2}{\parallel \nu \parallel }_{\mathrm{0,}\Omega }^{2}\ge {C}_{0}^{-2}({\parallel \lambda \parallel }_{\infty ,\Omega }-{\parallel h(t)\parallel }_{\infty ,{\Gamma }_{3}}{C}_{3}^{2}){\parallel \nu \parallel }_{\mathrm{0,}\Omega }^{2}\\ \forall \nu \in V\Rightarrow a(t;\nu ,\nu )+\underset{>0}{\underset{\underbrace{}}{\beta }}{\parallel \nu \parallel }_{\mathrm{0,}\Omega }^{2}\ge \underset{\alpha >0}{\underset{\underbrace{}}{\left\{\frac{\beta }{2}+{C}_{0}^{-2}({\parallel \lambda \parallel }_{\infty ,\Omega }-{\parallel h(t)\parallel }_{\infty ,{\Gamma }_{3}}{C}_{3}^{2})\right\}}}{\parallel \nu \parallel }_{\mathrm{0,}\Omega }^{2}\end{array}\)
with \({C}_{0}\) the continuity constant for the canonical injection of \({H}^{1}(O)\) into \({L}^{2}(O)\).
The continuity of the linear form \(b(t)\) on \(V\)
\(\begin{array}{}\text{pp}t\in ]\mathrm{0,}\tau [\mid (b(t),\nu )\mid \le {\parallel \stackrel{ˆ}{s}(t)\parallel }_{-\mathrm{1,}\Omega }{\parallel \nu \parallel }_{\mathrm{1,}{\Omega }_{3}}+{\parallel \stackrel{ˆ}{g}(t)\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}{\parallel \nu \parallel }_{\frac{1}{2},{\Gamma }_{2}}+{\parallel \stackrel{ˆ}{h}(t)\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}{\parallel \nu \parallel }_{\frac{1}{2},{\Gamma }_{3}}\\ \forall \nu \in V\le P(\Omega )\text{max}({\parallel \stackrel{ˆ}{s}(t)\parallel }_{\text{-}\mathrm{1,}\Omega },{\parallel \stackrel{ˆ}{g}(t)\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{2}}{C}_{2},{\parallel \stackrel{ˆ}{h}(t)\parallel }_{\text{-}\frac{1}{2},{\Gamma }_{3}}{C}_{3}){\mid \nu \mid }_{\mathrm{1,}\Omega }\end{array}\)
with \({C}_{2}\) the trace operator’s continuity constant on \({\Gamma }_{2}\).
Theorem 4
The problems \(({P}_{0})\) and \(({P}_{2})\) are equivalent and therefore the initial problem admits one solution and only one
\(u\in {L}^{2}(\mathrm{0,}\tau ;V)\cap {C}^{0}(\mathrm{0,}\tau ;H)\).
Proof:
The existence and the uniqueness of the solution of problem \(({P}_{0})\) of course results from the previous theorem, once the equivalence of the two problems has been demonstrated. It therefore remains to prove the opposite involvement \(({P}_{2})\Rightarrow ({P}_{0})\), which is very hard to exhume « not formally ». In particular, the Neumann and Robin limit conditions and the Cauchy condition are difficult to obtain rigorously. The « Dautray‑Lions » offers a very technical demonstration ([bib1] § XVIII pp637-641). By adapting these results we show that in our case, the boundary conditions on \({\Gamma }_{i}\) are in fact verified, not on \({L}^{2}(\mathrm{0,}\tau ,{H}^{\text{-}\frac{1}{2}}({\Gamma }_{i}))\), but on the space \(({B}_{i})\text{'}\supset {H}_{\text{00}}^{\text{-}\frac{1}{2}}({\Gamma }_{\tau }^{i})\) (by noting \({\Gamma }_{\tau }^{i}:={\Gamma }_{i}\times ]\mathrm{0,}\tau [\)) defined as being the topological dual of
\({B}_{i}:=\left\{w\in {H}^{\frac{1}{2}}(\partial {\Omega }_{\tau })\cap {L}^{2}({\Gamma }_{\tau }^{i})/\exists \nu \in {L}^{2}(\mathrm{0,}\tau ;V)\text{avec}{\nu }_{\mid \Omega \times \left\{0\right\}}={\nu }_{\mid \Omega \times \left\{\tau \right\}}=0\text{et}{\nu }_{\mid {\Gamma }_{i}}=w\right\}\)
Notes:
Now that we have ensured the existence and uniqueness of the solution within the functional framework required by the Code_Aster operators, we will semi-discretize in time ( \({P}_{0}\) ) **** **** **** and then spatially discretize everything using a finite element method. At the same time, we will study its stability properties. They will be very useful for identifying the norms, techniques and inequalities that will be involved in the genesis of the residual error indicator.