Please enable JavaScript.

Coggle requires JavaScript to display documents.

Sobolev Spaces (Ch3) (Basics (Ch3.1) (Derivatives (generalized derivative…

- - - - conditions for \( u \in H^1(\Omega)\)(Th46)
      - Construction of sub-domains \( \Omega_i \) and interfaces \( \gamma_{ij} \) in \( \Omega \) (P31)
      - \( \Rightarrow \) finite element space is sub-space of \( H^1 \)
    - - extension operator (eq3.3)
        \( (Eu)(\widehat{x})=u(x) \forall x \in \cup S_i \)
        \( (Eu)(x)=u(x) \forall x \in \Omega \)
      - Extension \( E_0u(\widehat{x})=(1-\eta) u(x) \) is extension from \( H^1(\Omega) \) to \( H^1_0(\widetilde{\Omega}) \) and its bounded (Th48) (\(E_0\) vanishes at \( \partial \widetilde{\Omega} \) )
      - Extension from \( H^1_0({\Omega}) \) onto larger domains trivial, by zero (P33)
      - Construction (P32)
      - ext. operator \( E:H^1(\Omega) \rightarrow H^1(\widetilde{\Omega}) \) is well defined and bounded (Th47)
  - - - interpolation space \( V_s \)
      - interpolation norm
      - goal: explicit form of the trace-norm