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Applications (Ch8) (Maxwell eq (Ch8.3) (setup in \( \mathbb{R} \) (current…

- - - - Dirichlet (displacement at boundary)
      - Neumann (stress at boundary )
  - - - vertical displacement \( w(x) \)
      - in principle well posed (Le104)
      - lin. rotation of normal vector \( \beta \)
      - small thickness
        \( \Rightarrow \) large cont./coerc. ratio
        \( \Rightarrow \) error increase
        
        mixed method
        
        extended Brezzi (Th105)
        
        idea: weaken term w/ large coefficient
      - average longitudinal displacing \( U(x) \)
      - \( \Rightarrow \) 2 decoupled problems
        
        longitdunial displacement
        
        bending & shearing
- - - - Elements w/ discont. pressure (Ch2.1)
        
        \( V_h = P^{1,nc}, \, Q_h = P^{0,dc} \)
        
        Fortin operator
        
        error-estimate
        
        convergence rate \( O(h) \) optimal
        
        solution = pairing
        \( V_h = P^{2+}, \, Q_h = P^{1,nc} \)
        
        2nd order space w/ cubic bubbles
        \( P^{2+}(\mathcal{T})=\{ v_h \in H^1:v_{h\vert T}\in P^3(T),v_{h\vert E} \in P^2(E) \} \)
        
        2nd order convergence
        
        cost low for add. bubbles
        
        \( V_h = P^{2}, \, Q_h = P^{0,dc} \)
        
        Fortin + Clement operator
        
        1st order convergence
        
        error-estimates
        
        correction operator
        
        this element sub-optimal
      - Elements w/ cont. pressure (Ch2.2)
        
        Fortin operator (preserving mean values)
        
        \( O(h) \) convergent
        
        clement operator
        
        mini element \( V_h = P^{1+}, \, Q_h = P^{1,cont} \)
        
        Alternative: Taylor-Hood element
        
        requires add. assumptions
        
        stable pair \( P^2 \times P^{1,cont} \)
        
        conv. rate \( O(h^2) \)
        
        correction operator
    - - stokes eq. = N-S-eq - viscos. & density terms
      - solvability by Brezzi
        
        Proof1: ignore Dirichlet and use extension operator
        
        Proof2: on star shaped \( \Omega \)
        
        LBB-cond. non-trivial
  - - - piece-wise constant for \( p \)
      - non-conf. \( P_1 \) for \( u \)
  - - - variables
        
        \( \Omega \in \mathbb{R} \)
        
        position \( x \in \Omega \)
        
        particle \( X \in \Omega\)
        
        velocity of particle \( u(x,t) \)
        
        evolution of particles \( \varphi : \Omega\times [0,T] \rightarrow \Omega \)
      - Langrangian
        
        solids
        
        coordinates \( X \)
      - Reynolds transport Theorem (Th1)
        
        \( \frac{d}{dt}\int_{V(t)} c(x,t)\mathrm{\,dx}= \int_{V(t)} \frac{\partial c}{\partial t}(x,t)\mathrm{\,dx}+ \int_{\partial V(t)} n\cdot uc\mathrm{\,ds} \)
        
        Interpretation
        
        quantity preserved
        
        material derivative
        \( \frac{Dc}{Dt} := \frac{\partial c}{\partial t} + \mathrm{div\,}(uc) \)
      - basis: conservation eq.s (e.g. mass)
      - Eulerian
        
        \( x \) variables
        
        fluids
    - - unknown vars
        
        vector valued velocity field \( u \)
        
        scalar-valued pressure field \( p \)
      - new var. form
      - conserved quants.
        
        momentum
        
        mass
    - - implicit + explicit Euler method (mixture used)
      - goal: rewrite terms to solve per time step
      - space-discretized system