From: frohne Date: Wed, 27 Feb 2013 23:14:56 +0000 (+0000) Subject: fixing space and set definitions in the saddle point formulation X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9834534f87ac6ff7ece169b0280ff2d79cee4288;p=dealii-svn.git fixing space and set definitions in the saddle point formulation git-svn-id: https://svn.dealii.org/trunk@28625 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-42/doc/intro-step-42.tex b/deal.II/examples/step-42/doc/intro-step-42.tex index 0a8e2d649e..17767acd14 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -185,9 +185,9 @@ method - inexact since we use an iterative solver for the linearised problems in \section{Linearization of the constitutive law for the Newton method} To apply Newton method we have to linearise the following semi-linearform -$$a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\varphi)),\varepsilon(\varphi)\right).$$ -Because we have to find the solution $u$ in the convex set $V^+$, we have to -apply an SQP-method (SQP: sequential quadratic programming). That means we have +$$a(\psi;\varphi) := +\left(P_{\Pi}(C\varepsilon(\psi)),\varepsilon(\varphi)\right).$$ Because we have +to find the solution $u$ in the convex set $V^+$, we have to apply an SQP-method (SQP: sequential quadratic programming). That means we have to solve a minimisation problem for a known $u^i$ in every SQP-step of the form \begin{eqnarray*} & & a(u^{i};u^{i+1} - u^i) + \dfrac{1}{2}a'(u^i;u^{i+1} - u^i,u^{i+1} - u^i)\\ @@ -229,14 +229,14 @@ Just as in step-41 we compose a saddle point problem out of the minimisation problem. Again we do so to gain a formulation that allows us to solve a linear system of equations finally.\\ We introduce a Lagrange multiplier $\lambda$ and the convex cone $K\subset W'$, -$W'$ dual space of the trace space $W$ of $V$ restricted to $\Gamma_C$, -$$K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad -\forall v\in W, v \ge 0\text{ on }\Gamma_C \}$$ +$W'$ dual space of the trace space $W:=\left[ H_0^{\frac{1}{2}}(\Gamma_C) +\right]^{\textrm{dim}$ of $V$ restricted to $\Gamma_C$, $$K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad +\forall v\in H_0^{\frac{1}{2}}(\Gamma_C), v \ge 0\text{ on }\Gamma_C \}$$ of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ denotes the duality pairing, i.e. a boundary integral, between $W'$ and $W$. Intuitively, $K$ is the cone of all "non-positive functions", except that $ K\subset -\left( \left[ H_0^{\frac{1}{2}} \right]^{\textrm{dim}} \right)' $ and so contains other -objects besides regular functions as well. This yields:\\ +\left( \left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}} \right)' $ and +so contains other objects besides regular functions as well. This yields:\\ \noindent \textit{Find $u\in V$ and $\lambda\in K$ such that} diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox index e90f986369..f33e5b31fd 100644 --- a/deal.II/examples/step-42/doc/intro.dox +++ b/deal.II/examples/step-42/doc/intro.dox @@ -198,7 +198,7 @@ method - inexact since we use an iterative solver for the linearised problems in

Linearisation of the constitutive law for the Newton method

For the Newton method we have to linearise the following semi-linearform -@f{gather*}a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\varphi)),\varepsilon(\varphi)\right).@f} +@f{gather*}a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\psi)),\varepsilon(\varphi)\right).@f} Because we have to find the solution $u$ in the convex set $V^+$, we have to apply an SQP-method (SQP: sequential quadratic programming). That means we have to solve a minimisation problem for a known $u^i$ in every SQP-step of the form @@ -244,13 +244,14 @@ problem. Again we do so to gain a formulation that allows us to solve a linear system of equations finally. We introduce a Lagrange multiplier $\lambda$ and the convex cone $K\subset W'$, -$W'$ dual space of the trace space $W$ of $V$ restricted to $\Gamma_C$, +$W'$ dual space of the trace space $W:=\left[ H_0^{\frac{1}{2}}(\Gamma_C) +\right]^{\textrm{dim}$ of $V$ restricted to $\Gamma_C$, @f{gather*}K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad -\forall v\in W, v \ge 0\text{ on }\Gamma_C \}@f} +\forall v\in H_0^{\frac{1}{2}}(\Gamma_C), v \ge 0\text{ on }\Gamma_C \}@f} of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ denotes the duality pairing, i.e. a boundary integral, between $W'$ and $W$. Intuitively, $K$ is the cone of all "non-positive functions", except that $ K\subset -\left( \left[ H_0^{\frac{1}{2}} \right]^{\textrm{dim}} \right)' $ and so contains other +\left( \left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}} \right)' $ and so contains other objects besides regular functions as well. This yields: Find $u\in V$ and $\lambda\in K$ such that