1
2
3\subsection{The special case of Newton's linearization of~(\ref{eq:toto1}) with FirstOrderType2R~(\ref{first-DS2})}
4
5
6Let us now proceed with the time discretization of~(\ref{eq:toto1}) with FirstOrderType2R~(\ref{first-DS2})  by a fully implicit scheme :
7\begin{equation}
8  \begin{array}{l}
9    \label{eq:mlcp2-toto1-DS2}
10     M x_{k+1} = M x_{k} +h\theta f(x_{k+1},t_{k+1})+h(1-\theta) f(x_k,t_k) + h \gamma r(t_{k+1})
11     + h(1-\gamma)r(t_k)  \\[2mm]
12     y_{k+1} =  h(t_{k+1},x_{k+1},\lambda _{k+1}) \\[2mm]
13     r_{k+1} = g(t_{k+1},\lambda_{k+1})\\[2mm]
14  \end{array}
15\end{equation}
16
17
18 \paragraph{Newton's linearization of the first line of~(\ref{eq:mlcp2-toto1-DS2})} The linearization of the first line of the  problem~(\ref{eq:mlcp2-toto1-DS2}) is similar to the previous case so that (\ref{eq:rfree-2}) is still valid.
19
20
21 \paragraph{Newton's linearization of the second  line of~(\ref{eq:mlcp2-toto1-DS2})} The linearization of the second line of the  problem~(\ref{eq:mlcp2-toto1-DS2}) is similar to the previous case so that (\ref{eq:NL11y}) is still valid.
22
23 \paragraph{Newton's linearization of the third  line of~(\ref{eq:mlcp2-toto1-DS2})}
24Since $ K^{\alpha}_{k+1} = \nabla_xg(t_{k+1},\lambda ^{\alpha}_{k+1}) = 0 $, the linearization of the third line of (\ref{eq:mlcp2-toto1-DS2}) reads as
25\begin{equation}
26  \label{eq:mlcp2-rrL}
27  \begin{array}{l}
28    \boxed{r^{\alpha+1}_{k+1} = g(t_{k+1},\lambda ^{\alpha}_{k+1})     + B^{\alpha}_{k+1} ( \lambda^{\alpha+1}-  \lambda^{\alpha}_{k+1} )}
29  \end{array}
30\end{equation}
31
32
33\paragraph{Reduction to a linear relation between  $x^{\alpha+1}_{k+1}$ and
34$\lambda^{\alpha+1}_{k+1}$}
35
36Inserting (\ref{eq:mlcp2-rrL}) into~(\ref{eq:rfree-11}), we get the following linear relation between $x^{\alpha+1}_{k+1}$ and
37$\lambda^{\alpha+1}_{k+1}$, we get the linear relation
38\begin{equation}
39  \label{eq:mlcp2-rfree-13}
40  \begin{array}{l}
41 \boxed{   x^{\alpha+1}_{k+1}\stackrel{\Delta}{=} x^\alpha_p + \left[ h \gamma (W^{\alpha}_{k+1})^{-1}    B^{\alpha}_{k+1} \lambda^{\alpha+1}_{k+1}\right]}
42   \end{array}
43\end{equation}
44with
45\begin{equation}
46  \boxed{x^\alpha_p \stackrel{\Delta}{=}  h\gamma(W^{\alpha}_{k+1} )^{-1}\left[g(t_{k+1},\lambda^{\alpha}_{k+1})
47    -B^{\alpha}_{k+1} (\lambda^{\alpha}_{k+1}) \right ] +x^\alpha_{\free}}
48\end{equation}
49and
50\begin{equation}
51  \label{eq:mlcp2-NL9}
52  \begin{array}{l}
53    W^{\alpha}_{k+1} \stackrel{\Delta}{=} M-h\theta A^{\alpha}_{k+1}\\
54  \end{array}
55\end{equation}
56
57
58\paragraph{Reduction to a linear relation between  $y^{\alpha+1}_{k+1}$ and
59$\lambda^{\alpha+1}_{k+1}$}
60
61Inserting (\ref{eq:mlcp2-rfree-13}) into (\ref{eq:NL11y}), we get the following linear relation between $y^{\alpha+1}_{k+1}$ and $\lambda^{\alpha+1}_{k+1}$,
62\begin{equation}
63   \begin{array}{l}
64 y^{\alpha+1}_{k+1} = y_p + \left[ h \gamma C^{\alpha}_{k+1} ( W^{\alpha}_{k+1})^{-1}  B^{\alpha}_{k+1} + D^{\alpha}_{k+1} \right]\lambda^{\alpha+1}_{k+1}
65   \end{array}
66\end{equation}
67with
68\begin{equation}\boxed{
69y_p = y^{\alpha}_{k+1} -\mathcal R^{\alpha}_{yk+1} + C^{\alpha}_{k+1}(x_q) -
70D^{\alpha}_{k+1} \lambda^{\alpha}_{k+1} }
71\end{equation}
72\textcolor{red}{
73  \begin{equation}
74    \boxed{ x^\alpha_q= x^\alpha_p - x^{\alpha}_{k+1}\label{eq:mlcp2-xqq}}
75  \end{equation}
76}
77
78\subsection{The special case of Newton's linearization of~(\ref{eq:toto1}) with FirstOrderType1R~(\ref{first-DS1})}
79
80
81Let us now proceed with the time discretization of~(\ref{eq:toto1}) with FirstOrderType1R~(\ref{first-DS1})  by a fully implicit scheme :
82\begin{equation}
83  \begin{array}{l}
84    \label{eq:mlcp3-toto1-DS1}
85     M x_{k+1} = M x_{k} +h\theta f(x_{k+1},t_{k+1})+h(1-\theta) f(x_k,t_k) + h \gamma r(t_{k+1})
86     + h(1-\gamma)r(t_k)  \\[2mm]
87     y_{k+1} =  h(t_{k+1},x_{k+1}) \\[2mm]
88     r_{k+1} = g(t_{k+1}\lambda_{k+1})\\[2mm]
89  \end{array}
90\end{equation}
91
92The previous derivation is valid with $ D^{\alpha}_{k+1} =0$.
93
94
95
96%%% Local Variables:
97%%% mode: latex
98%%% TeX-master: "DevNotes"
99%%% End: