Please enable JavaScript.

Coggle requires JavaScript to display documents.

FEM (Ch5) (Finite element error analysis (Ch5.2) (\( h_T \)diameter of…

- - - - difficult: interpolation operator to an \( H^1 \) conforming FE space
      - easy
        
        on n-dim simplices
        
        in 1D
        
        on tensor product elements
    - - projection based interp.
      - \( I_p\)err estimates for \( u \in H^1, p \in L_2, q \in \Pi_p \)
      - p-version Galerkin Method for Dirichlet problem through operator \( I_p\)
      - commuting diagram \( \Pi_{L_2}^{\Pi_{p-1}}u'=(I_pu)' \) (Le6)
    - - \( \exists \) extension \( \widetilde{v}\in \Pi_p(T) \) (Le7)
      - err estimate (which was the goal) (Le8)
- - - - usual spaces for \( T \subset \mathbb{R}^2\)
        
        \( Q^p \)
        
        \( P^p \)
      - nodal basis \( \{\varphi_T^i \}\) for \( V_T \), dual to \( \Psi_T\)
      - local nodal interpolation operator \( I_T v\) (P46) (is projection)
    - - point evaluations functionals
      - are equivalent
    - - linear or quadratic line segment
      - Hermite line segment
      - triangle
        
        Morely
        
        constant
        
        non-conforming
        
        linear
        
        Raviart-Thomas
    - - interpolation equivalent
      - not equivalent
      - using point eval. and derivatives
- - - - computation of the lifting \( \Vert \sigma^{\Delta}\Vert \) (Ch2)
        
        explicit construction
        
        proves existence of equilibrated flux
        
        of equilibrated fluxes on vertex patch
        
        of \( \sigma^{\Delta} \) in terms of BDM elements
        
        instead: solve local optimization problem
        
        rewrite \( r() \) as functional
        
        localize construction of flux
        
        further notes/attrinbutes
        
        applies also in 3D
      - framework (Ch1)
        
        residual \( r(\cdot)\in V^*\)
        \(r(v)=f(v)-A(u_h,v),\,v\in V\)
        
        res-err estimator w/ weighted \(L_2\)-norms
        
        name: flux-postprocessing
        \( \Rightarrow \textrm{div}\sigma=f \Rightarrow \) flux in exact equlibrium with source \( f \)
        
        different representation by sums of
        
        edge-res (normal jump) \( r_E \)
        
        element-res \( r_T \)
        
        problem: direct eval. of \( \Vert r \Vert\) not feasible
        
        equlibrated res err estimator \( \eta^{er}\)
        
        efficient w/ generic constant c
        
        reliable w/ c=1
        
        goal: provide upper-bounds for discretization err w/o generic constants
        
        main idea: calculate lifting \( \sigma^{\Delta}\)
        
        evaluation \(L_2\) norm of \( \sigma^{\Delta}\) easy
        
        such that \( \textrm{div}\sigma^{\Delta}=r\)