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: