Aufgaben in gemischter Formulierung

Allgemein:

$\mathbf{u}\in X$, $p \in M$
\begin{displaymath}\begin{array}{rclcl@{\qquad}l} a(\mathbf{u},\mathbf{v}) & + &... ...,\mathbf{u}) & - & c(p,q) & = & 0 & \forall q\in M \end{array} \end{displaymath}

Elastizität:

$\mathbf{u}\in H^1_0(\Omega)^d$, $p \in L^2(\Omega)$
\begin{displaymath}\begin{array}{rclcl} (2\mu\,\varepsilon(\mathbf{u}), \varepsi... ...\,}}\mathbf{u}) & - & \lambda^{-1}\,(p,q) & = & 0 \end{array} \end{displaymath}

Stokes:

$\mathbf{u}\in H^1_0(\Omega)^d$, $p \in L^2_0(\Omega)$
\begin{displaymath}\begin{array}{rclcl} (\nabla\mathbf{u},\nabla\mathbf{v}) & + ... ... (q, \mbox{\textbf{div\,}}\mathbf{u}) & & & = & 0 \end{array} \end{displaymath}

Reissner-Mindlin:

$(w,\mathbf{\beta})\in H^1_0(\Omega)^3$, $\mathbf{\gamma}\in L^2(\Omega)^2$
\begin{displaymath}\begin{array}{rclcl} a(\mathbf{\beta},\mathbf{\psi}) & + & (\... ... & t^{-2} (\mathbf{\gamma},\mathbf{\eta}) & = & 0 \end{array} \end{displaymath}

mit
\begin{displaymath}a(\mathbf{\beta},\mathbf{\psi}) = (2\mu\,\varepsilon(\mathbf... ...xtbf{div\,}}\mathbf{\beta},\mbox{\textbf{div\,}}\mathbf{\psi}) \end{displaymath}

Für MITC-Diskretisierung:

Ansatz:
\begin{displaymath}\mathbf{\gamma}=\nabla r + \mbox{\textbf{curl\,}}p \qquad \mathbf{\eta} = \nabla z + \mbox{\textbf{curl\,}}q \end{displaymath}

$\Downarrow$
  1. $r\in H^1_0(\Omega)$
    \begin{displaymath}(\nabla r,\nabla v) = \langle g,v \rangle \end{displaymath}

  2. $\mathbf{\beta}\in H^1_0(\Omega)^2$, $p\in H^1(\Omega)/\mathrm{R}$
    \begin{displaymath}\begin{array}{rclcl} a(\mathbf{\beta},\mathbf{\psi}) & - & (p... ...url\,}}p,\mbox{\textbf{curl\,}}q) & = & 0 \\ [1ex] \end{array} \end{displaymath}

  3. $w\in H^1_0(\Omega)$
    \begin{displaymath}(\nabla w,\nabla z) = (\mathbf{\beta}, \nabla z) + t^2 \langle g,z \rangle \end{displaymath}

Alexander Smuglyakov