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NUMPDE (- [X] abstract theory (Ch2) (Inf-Sup stable variations (Ch2.6)…

- - - - normed (Def2)
        
        complete = Banach Space (BS) (Def3)
        
        closure \( \overline{W}^{\left\Vert \cdot\right\Vert _{V}} \) (Def4)
      - functional = linear form (LF) (Def5)
        
        bounded (Def5)
        
        dual space \( V^* \) (Def5)
        
        Notation: duality product \( \left\langle f,v\right\rangle_{V^* \times V} =f\left(v\right) \) (P18)
        
        dual norm (Def5)
      - bilinear form (BLF)(Def6)
        
        symmetric
        
        inner product \( (\cdot , \cdot)_V \) (IP) (Def7)
        
        inner product space \( \left(V,(\cdot,\cdot)_{V}\right) \) (Def8)
        
        \( \left\Vert v\right\Vert _{V}:=\sqrt{\left(v,v\right)_{V}} \) is induced norm on \( \left(V,\left(\cdot,\cdot\right)_{V}\right) \)
        
        cont. BLF induces cont. lin. operator (Le28)
        \(A:V \rightarrow V^* \) via \( \left\langle Au,v\right\rangle=A\left(u,v\right) \)
      - Hilbert space (HS)(Def11)
        
        closed subspace (Def12)
        
        S subspace of V \( \Rightarrow S^\perp = \left\{ v \in V: (v,w)=0 \forall w \in S\right\} \) is closed subspace of V
        
        finite dim. subspace is closed subspace
        
        \( \ker T:=\left\{ v\in V:Tv=0 \right\} \) is closed subspace of HS V, where T is cont. lin. operator (Le13)
      - linear operator(Def15)
        
        bounded (Def15)
        
        compact (Def19)
        
        operator compact \( \Leftrightarrow \exists \) Eigensystem of the operator (Le20)
        
        extension principle for operators (Le18)
        
        \( \Rightarrow \) continous (Le16)
        
        operator norm(Def15)
      - dense subspace (Def17)
  - - - \( \Rightarrow \) unique decomposition of \( u \in V \)
        \( u=u_0 + u_1 \)
        \( u_0\in S , u_1 \in S^\perp \)
      - \( \Rightarrow \) define operators \( P_S, P_S^\perp \)
        
        linear (Th22)
        
        orthogonal (Th24)
    - - orthogonal \( \left(Pu,v \right) = \left(u,Pv \right) \) (Def23)
  - - - \( ( C^1(\Omega), \left\Vert \cdot \right\Vert _{A})\) not complete
      - \( V=\overline{C^{1}(\Omega)}^{\left\Vert \cdot\right\Vert _{A}} \) is HS
    - - non-neg
      - \( A(u,u)=0 \Rightarrow u=0\)
      - \( \Rightarrow A(\cdot,\cdot) \) is IP and \( \left\Vert \cdot\right\Vert _{A} \) is norm
    - - if bounded \( \Rightarrow \) cont.
      - with Riesz \( \Rightarrow \exists!u:A\left(u,v\right)=f\left(v\right)\Rightarrow \) weak form has unique solution
    - - \( dim(V_h)<\infty \Rightarrow closed\)
      - \( u_h \) is projection of \( u \) on \( V_h \)
      - error \( u-u_h \) is orth. to \( V_h \)
  - - - BLF coercive aka elliptic & cont.
      - LF cont
      - \(V\) is HS
      - rewrite var problem as operator eq. (P19)
        Find \(u\) that \( Au=f \) in \( V^*\)
    - - Diffusion-reaction eq. (Ex26)
      - Diffusion-convection-reaction eq. (Ex27)
    - - unique solvability (inherited from original problem)
      - Cea - error is quasi optimal (Th31)
      - if BLF A is symmetric \( \Rightarrow \) factor in error-estimate is sqrt of Th31
  - - - \( \Rightarrow B\) is one-to-one (=injective)
    - - coercive BLF (Ex34)
      - complex symmetric var. problem (Ex35)
    - - linear operator \(B:V \rightarrow W^* \) by \( \left\langle Bu,v \right\rangle_{W^*\times W}=B(u,v) \) as in Le28 (P22)
    - - \( \Rightarrow \) operator B has closed range (Le32)
      - & weaker condition is fulfilled \( \Rightarrow \) problem has unique solution, which depends cont. on right side (Le33)
    - - quasi optimal error estimate (Th36)
      - solvability of finite dim eq does not follow from solvability of original problem --> extra inf-sup condition for discrete problem
- - - - Motivation: optimal balance of mesh size \(h \) & polynomial order \( p \) to limit better than \( \inf_{v_{hp}\in V_{hp}} \Vert u-v_{hp} \Vert_{H^1}\leq ch^{m-1}\Vert u \Vert_{H^m} \)
      - goal: p-version err estimate \(\leq c(\frac{h}{p})^{m-1}\Vert u \Vert_{H^m} \)
      - hp-version would lead to exp. conv. \( \leq ce^{-N^{\alpha}}\)
    - - Def via Rodrigues formula
        \( P_n (x) := \frac{1}{2^nn!} \frac{d^n}{dx^n} (x^2-1)^n\)
      - \( \int\limits^{1}_{-1}P_n(x)\,P_m(x)\, \textrm{dx}=\frac{2}{2n+1}\delta_{n,m}\) (Le1)
      - formula \( (n+1)P_{n+1}(x) \) (Le2)
      - Sturm-Liouville diff. eq. satisfied (Le3)
      - orthogonal w.r.t \( (u',v')_{L_2,1-x^2}\) as implication of (Le3)
    - - polynomials dense in \(L_2(-1,1) \Rightarrow u=\sum\limits_{n=0}^{\infty}a_nP_n\)
        with \( a_n \) generalized Fourier coeffs.
      - \( P_{L_2}^{\Pi_p} \) is \( L_2 \) projection to \({\Pi_p} \)
      - projection err is bounded (Le4)
    - - (JP) Jacobi Polynomials (P5)
      - JP are orthogonal w.r.t. the weighted IP
      - Dubiner basis = \( L_2(T) \) orthogonal basis for \( \Pi_p(T)\) by JPs(Le5)
    - - approx. err estimates of orth. polynomials (Ch4.0)
        
        easy
        
        on tensor product elements
        
        on n-dim simplices
        
        in 1D
        
        difficult: interpolation operator to an \( H^1 \) conforming FE space
      - 1D-case (Ch4.1)
        
        p-version Galerkin Method for Dirichlet problem through operator \( I_p\)
        
        projection based interp.
        
        commuting diagram \( \Pi_{L_2}^{\Pi_{p-1}}u'=(I_pu)' \) (Le6)
        
        \( I_p\)err estimates for \( u \in H^1, p \in L_2, q \in \Pi_p \)
      - ...on triangles (Ch4.2)
        
        \( \exists \) extension \( \widetilde{v}\in \Pi_p(T) \) (Le7)
        
        err estimate (which was the goal) (Le8)
  - - - Equilibrated Residual Error Estimates (Supp)
        
        framework (Ch1)
        
        goal: provide upper-bounds for discretization err w/o generic constants
        
        residual \( r(\cdot)\in V^*\)
        \(r(v)=f(v)-A(u_h,v),\,v\in V\)
        
        different representation by sums of
        
        element-res \( r_T \)
        
        edge-res (normal jump) \( r_E \)
        
        res-err estimator w/ weighted \(L_2\)-norms
        
        equlibrated res err estimator \( \eta^{er}\)
        
        reliable w/ c=1
        
        efficient w/ generic constant c
        
        problem: direct eval. of \( \Vert r \Vert\) not feasible
        
        name: flux-postprocessing
        \( \Rightarrow \textrm{div}\sigma=f \Rightarrow \) flux in exact equlibrium with source \( f \)
        
        main idea: calculate lifting \( \sigma^{\Delta}\)
        
        such that \( \textrm{div}\sigma^{\Delta}=r\)
        
        evaluation \(L_2\) norm of \( \sigma^{\Delta}\) easy
        
        computation of the lifting \( \Vert \sigma^{\Delta}\Vert \) (Ch2)
        
        rewrite \( r() \) as functional
        
        localize construction of flux
        
        explicit construction
        
        of equilibrated fluxes on vertex patch
        
        of \( \sigma^{\Delta} \) in terms of BDM elements
        
        proves existence of equilibrated flux
        
        instead: solve local optimization problem
        
        applies also in 3D
        
        further notes/attrinbutes
      - reliability (Th71) + Proof
      - Clement quasi-interpolation operator (Def69)
    - - reliable \( \left\Vert u-u_h \right\Vert_V \leq C_1 \eta (u_h, f) \)
      - efficient \( \left\Vert u-u_h \right\Vert_V \geq C_2 \eta (u_h, f) \)
      - usual error estimators \( \eta^2 (u_h, f) = \sum_{T \in \mathcal{T}} \eta^2_T (u_h, f) \)
      - local efficiency estimates \( \left\Vert u-u_h \right\Vert_{H^1(\omega_T)} \geq C_2 \eta_T (u_h, f) \)
    - - pseudo code
      - Red Green Refinement
      - marked edge bisection
  - - - \( T \) bounded set
      - \( V_T \) function space on \( T \)
        
        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)
      - \( \Psi_T \) lin. independent functionals on \( V_T \)
      - Examples
        
        linear or quadratic line segment
        
        Hermite line segment
        
        triangle
        
        constant
        
        linear
        
        non-conforming
        
        Morely
        
        Raviart-Thomas
      - Lagrange finite elements
        
        point evaluations functionals
        
        are equivalent
      - Hermite finite elements
        
        using point eval. and derivatives
        
        not equivalent
        
        interpolation equivalent
    - - \( C^m \) smooth functions
      - regularity (P47)
      - Examples: \( P^1, P^0\) with different node locations
    - - reference finite element (P46)
      - affine equivalent
      - interpolation equivalent
      - equivalent \( \Rightarrow \) interpolation equivalent (Le60)
    - - global interpolation operator \( I_{\mathcal{T}}v_{\vert T} \) (P47)
      - (regular) triangulation \( \mathcal{T}=\{ T_1,...,T_M \} \) (P46)
      - connectivity matrix \( C_T \in \mathbb{R}^{N \times N_T} \) (P47)
  - - - finite element problem
      - obtain linear system
      - numerical integration
    - - inserting results in reduced system
      - or approx. by Robin BC
  - - - shape regular (for families of triangulations)
      - quasi uniform triangulation/mesh (P51)
    - - dual problem
      - Aubin-Nitsche (Th66) + Proof
        \( \left\Vert u-u_h \right\Vert_{L_2} \preceq h^2\left\Vert u \right\Vert_{H^2} \)
  - - - Assume: coerc. & cont. of \( A_h(.,.) \)
      - \( \Vert u-u_h \Vert_h \preceq \inf_{v_h \in V_h} \Vert u-v_h \Vert_h + \sup_{w_h \in V_h} \frac{A_h(u,w_h)-f_h/w_h)}{\Vert w_h \Vert_h}\)
      - BLF & LF replaced by mesh dependent forms
      - no requirement for \( V_h \subset V \)
    - - \( V_h^{nc}=\left\{ v \in L_2 : v_{\vert T} \in P^1 (T), v \mathrm{\,is\,cont.\, in\, edge\,midpoints} \right\}\)
      - 2nd Lemma of Strang gives error estimate \( \left\Vert u-u_h \right\Vert \preceq h \Vert u \Vert_{H^2} \)
      - Applications (NS-eq, \(L_2\) Matrix is diag.)
- - - - Primal & dual mixed formulations (Ch1)
        
        primal form
        
        standard \( H^1 \) finite element \(u\)
        
        discont. \( L_2 \) elements \(\sigma\)
        
        dual mixed form
        
        RT- or BDM-elements \(\sigma\)
        
        \( L_2\)-elements \(u\)
        
        locally exact conservation \( \int_{\partial T}\sigma_n =- \int_T f \)
        
        satisfies Brezzi's condition (Th1)
        
        discrete \( H^1 \)-norm or DG-norm
      - super-convergence of the scalar (Ch2)
        
        err estimate for RT-elements & \(u\) (Th2)
        applies also on BDM-elements + err estimates for both(Re3)
        
        super-convergence (Re4)
        
        local post-processing for scalar part
      - Solution methods for the lin. system (Ch3)
        
        discretization \( \Rightarrow \) lin. system=matrix
        
        is indefinite with
        
        \( \dim(V_h)\) pos. EVa
        
        \( \dim(Q_h)\) neg. EVa
        
        \( \Rightarrow \) difficult for lin. eq. solvers
        
        possibilities (none is good)
        
        direct solver (must apply Pivot strategies)
        
        block elimination
        not feasible, because \( A^{-1} \) not sparse
        
        extension of conjugate gradient method
        precond. important
      - Hybridization (Ch4)
        
        alternative technique/framwork
        
        new var. form = hybrid problem
        
        extra eq.s & \( \widehat{u} \) vars on element edges
        
        discrete system well posed
        
        solutions correspond to each other
        
        \(A\) block diagonal \( \Rightarrow\) cheap Schur complement
  - - - two problems
      - can be written as one by addition
      - and as operator eq.
        Riesz delivers operators \( A,B,B^* \)
      - \( V_0 := \{v:Bv=0\} \)
    - - LBB condition:
        \( \sup_{u \in V}\frac{b(u,q)}{\Vert u \Vert_V} \geq \beta_1 \Vert q \Vert_Q, \, \forall q \in Q\)
      - \( a(.,.) \) and \( b(.,.) \) are cont. BLFs
      - \( a(.,.) \) coercive on the kernel \( V_0 \)
      - \( \Rightarrow \) uniquely solvable and fulfills stability estimate
        \( \Vert u \Vert_V + \Vert p \Vert_Q \leq c(\Vert f \Vert_{V^*} + \Vert g \Vert_{Q^*} ) \)
  - - - problem form (spaces)
        \( u\in V=H^1, \, \lambda \in Q=H^{-1/2}(\Gamma_D) \)
      - has unique solution (Th87)
    - - \(g \) weak divergence of \( \sigma \) if (Def88)
        \( \int_{\Omega} g \varphi\,\mathrm{dx}=-\int_{\Omega} \sigma\cdot\nabla\varphi\,\mathrm{dx}, \,\forall\varphi\in C_0^\infty (\Omega) \)
      - \( H(\mathrm{div}, \Omega) := \{ \sigma \in [L_2(\Omega)]^d:\mathrm{div}\,\sigma\in L_2\}\)
        \( \Vert \sigma \Vert_{H(\mathrm{div})}=\sqrt{\Vert \sigma \Vert_{L_2}^2 + \Vert \mathrm{div}\sigma \Vert_{L_2}^2} \) (Def88)
      - problem form (spaces, BLFs,...)
        \( \sigma\in V=H(\mathrm{div}), \, u \in Q=L_2 \)
      - problem is well posed (Th89)
    - - normal-trace operator (Th90)
        \( \mathrm{tr}_n :H(\mathrm{div}) \rightarrow H^{-1/2}(\partial \Omega) \)
        for \( \sigma\in H(\mathrm{div}) \cap [C(\overline{\Omega})]^d \) holds \( \mathrm{tr}_n\sigma = \sigma\cdot n,\,on\,\partial\Omega \)
      - integration by parts (Le91)
      - BDM-elements (P104)
        \( V_h=\{ \sigma \in [L_2]^2:\sigma\vert_T\in[P^k]^d,\sigma\cdot n \mathrm{\,cont. across\,edges} \} \)
      - Raviart Thomas element
        
        global finite element function \(\sigma\) defined by
        \( \sigma\vert_T\in V_T \) and \( \int_{e_i}\sigma\vert_T \cdot n_{e_i}\mathrm{\,ds}=\sigma_i \)
        \(\forall e_i \subset T\) and \( \forall T \in \mathcal{T} \)
        
        RT-interpolation operator (Le93)
        \( I_h^{RT} \sigma = \sum_\limits{e_i}( \int_{e_i}\sigma\cdot n_{e_i} \mathrm{ds})\varphi_i^{RT} \)
        satisfies
        \( \int_T \mathrm{div\,} I_h\sigma\mathrm{\,dx}= \int_T \mathrm{div\,} \sigma\mathrm{\,dx},\,\forall T \in \mathcal{T}\)
        
        lowest order RT (=RT0-)element (cheapest)(P104)
        \( V_T = \{ \sigma=\left(\begin{array}{c}a\\b\end{array}\right)+c\left(\begin{array}{c}x\\y\end{array}\right):a,b,c \in \mathbb{R}\} \)
        \( \psi_i(\sigma)=\int_{e_i}\sigma\cdot n\mathrm{\,ds},\, i=1,2,3\)
      - Piola Transformation (Def94+Le95)
      - commuting diagram property with \( I_h \) & projection \( P_h\) (P105)
      - RT-triangle & RT-ref.triangle are interp. equivalent (Le96)
      - RT-Interpolation operator satisfies the approx. properties (Th97)
  - - - discrete mixed problem
      - Discrete stability = the discrete mixed problem is well posed(Le100)
      - A priori (error) estimate (Th101)
    - - formulation
      - well posed (Th102)
- - - - Setup
        \( \frac{\partial u}{\partial t}+ \textrm{div}\, f(u)=0 \)
      - Examples
        
        Burgers' eq
        \( f(u) = \frac{1}{2} u^2 \)
        
        Wave eq.
        
        Euler eq.
        
        Transport eq.
    - - weak solutions
        & Rankine-Hugoniot relation (Ch1.1)
        
        shock=
        intersection of characteristic lines
        
        speed = RH-relation \( s' \)
        \( s'=\frac{f(u_l)-f(u_r)}{u_l-u_r} \)
        
        position \( s(t) \)
        
        Ex: Burgers'
        \( s'=\frac{u_l+u_r}{2} \)
        
        weak solution in space-time
      - Expansion fans(Ch1.2)
        pick the meaningful physical solution
        
        Viscosity solutions \( \lim_{\epsilon \rightarrow 0}u^{\epsilon} \)
        \( u^{\epsilon}_t+f(u^{\epsilon})_x -\epsilon u^{\epsilon}_{xx}=0\)
        
        Entropy solutions
        
        Ex: Burgers'
    - - Galerkin methods
        (=natural ones)
        
        finite volume
        
        Discontinuous
      - up-wind like fluxes
      - entropy-viscosity methods
      - space-time methods
- - - - wind \( b\)
      - source \( f\)
      - BCs (inflow /outflow)
    - - considered as stationary transport in space-time
        \( \textrm{div}_{x,t}(\widetilde{b}u)=f \) on \( \Omega \)
      - conservation principle (outflow = production inside)
      - var. form.
      - inflow-outflow isometrie
        for \(f=0\) and \(v=u\)
        
        stability argument
        
        ensures conservation of \( L_2 \) norm in time
    - - var problem
      - space \(V\) by (semi-)norm
        \( \Vert u \Vert_V = \Vert b\nabla u \Vert_{L_2} \)
        
        vortices in \( b \)
        
        space-time problems -> no vortices
      - test space \( W=L_2 \)
      - BLF & LF cont. + inf-sup cond. trivial
  - - - cont. case disappears
      - discont. case important
  - - - Dirichlet BC in weak sense and obtain pos.def. matrix
      - Nitsche's Method (BLF, LF, norm)
      - BLF elliptic on FE space (Le2)
      - err ana
        
        apply discrete stability & consistency
        
        estimate interp. err in all terms of A
        
        not able to argue with cont. of \( A \) (because not cont.)
      - add consistent & stability terms (1st for symmetric BLF)
    - - var. form.
        
        consistent to solution
        
        elliptic on \( V_h \)
      - mean value of normal derivative & jump
      - add consistent. & stabilization(as before)
      - alternative to MM (Mortar)
      - leads to pos.-def. matrices (gluing method)
  - - - var. problem consistent to solution
      - BLF elliptic in any case \( \alpha > 0 \)
      - not consistent for dual problem --> Aubin-Nitsche not applicable
    - - disadvantages of DG
        
        higher # dofs
        
        higher # (non-zero terms per row in Matrix) --> HDG
      - derivation
        
        consistent terms for symmetric. & coerc.
        
        natural norm
        
        add variables \( \widehat{u},\widehat{v} \) on inter-element facets
      - convergence \( O(h) \)
      - Lehrenfeld-Trick
        
        projector
        
        (order-edges)-1 with same order of convergence
    - - disadvantage of IP-DG: Penalty term w/ \( \alpha \)
      - replace penalty term by BR norm
- - - - IC
      - linear/non-linear operators
    - - sub-problems
      - combine existing solvers
    - - bounded \( \Rightarrow \) convergent series
        \( e^A=\sum\limits^{\infty}_{n=0} \frac{1}{n!}A^n\)
      - unbounded \( \Rightarrow \) exp. is solution of diff. eq.
      - \( A \) elliptic \( \Rightarrow e^{-tA}\) bounded operator
  - - - left-sided rectangular rule = 1 step expl. Euler + solve simpler ODE
      - right-sided rectangular rule = solve A-ode + 1 step of implicit Euler
      - implicit midpoint rule = combi
- - - - compact \( \Leftrightarrow \) bounded
      - if \( \Omega \) is bounded domain: \( supp\{u\} \) is compact in \( \Omega \Leftrightarrow u\) vanishes on \( \partial\Omega \)
    - - generalized derivative \( D^\alpha_g u \) of \(u \in \mathcal{D}' \)(Def37)
      - weak derivative \( D^\alpha_w u \) of \(u \in L^{loc}_1(\Omega)\)(Def38)
      - weak is more general than classical, but more restrictive than generalized
  - - - well defined & cont. trace operator \( tr:H^1((0,h)) \rightarrow \mathbb{R} \) (Th44)
      - well defined & cont. trace operator \( tr:H^1(\Omega) \rightarrow L_2(\partial\Omega) \) (Th45)
    - - integration by parts in \( H^1 \) (P31)
      - Sobolev spaces over sub-domains (P31)
        
        Construction of sub-domains \( \Omega_i \) and interfaces \( \gamma_{ij} \) in \( \Omega \) (P31)
        
        conditions for \( u \in H^1(\Omega)\)(Th46)
        
        \( \Rightarrow \) finite element space is sub-space of \( H^1 \)
      - Extension operators (P32)
        
        Construction (P32)
        
        extension operator (eq3.3)
        \( (Eu)(\widehat{x})=u(x) \forall x \in \cup S_i \)
        \( (Eu)(x)=u(x) \forall x \in \Omega \)
        
        ext. operator \( E:H^1(\Omega) \rightarrow H^1(\widetilde{\Omega}) \) is well defined and bounded (Th47)
        
        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)
    - - space & norm (P34 & eq3.4)
      - is BS and existence of corresponding function in \( H^1 \) + Proof (Le49) = Trace Theorem?
      - Hilbert space interpolation (P35)
        
        interpolation norm
        
        interpolation space \( V_s \)
        
        goal: explicit form of the trace-norm
      - trace space onto one edge \( E \) = interpolation space \( H^{1/2}(E) \)
  - - - Application: \( V=H^1, W=L_2, \left\Vert v\right\Vert_A = \left\Vert \nabla v\right\Vert_{L_{2}} \)
    - - \( \Rightarrow \) compact embedding \(id: H^1 \rightarrow H^0=L_2 \)
    - - \( \Rightarrow \left\Vert u\right\Vert _{H^1(\Omega)}^2 \simeq \left\Vert \nabla v\right\Vert _{L_{2}(\Omega)}^2 + \left\Vert v\right\Vert _{L_{2}(\omega)}\), with \( \omega \subset \Omega, \left\vert \omega \right\vert > 0 \) in \( \mathbb{R}^d \)
      - \( \Rightarrow \left\Vert u\right\Vert _{H^1(\Omega)}^2 \simeq \left\Vert \nabla v\right\Vert _{L_{2}(\Omega)}^2 + \left\Vert v\right\Vert _{L_{2}(\gamma)}\), with \( \gamma \subset \partial\Omega, \left\vert \gamma \right\vert > 0 \) in \( \mathbb{R}^{d-1} \)
- - - - Family of uniformly cont. & elliptic operators
        \( A(t):V \rightarrow V^* \)
      - parabolic eq. + var. form
      - adding up generates new var. problem
      - is uniquely solvable (Th1=Lions)
    - - discretize in time
      - spatial function space dim= \( \infty \)
      - eq. for time-step
        = trapezoidal method (Crank Nicolson)
        A- but not L-stable
      - cannot prove discrete inf-sup cond.
    - - Integration by parts in time = time-slabs
      - plugin IC \( u_0 \Rightarrow IC \) part of var. form
      - give meaning to \(u(T)\)
        
        new variable for \(u(T)\)
        
        restrict test-space
        
        both well posed
      - parabolic eq. for every time-slab
        
        Initial value is end of previous time-slab
        
        first mesh \( \mathcal{T}\)
        
        2nd mesh-dependent form.
      - \( X_h=Y_h=P^{k,dc}(V) \)
      - mesh dependent norms
      - implicit Euler
        
        A- stable
        
        L-stable
      - BLF well defined & cont. on \( X_h\times Y_h \)
      - discrete problem is inf-sup stable on \( X_h\times Y_h \)
      - discrete err w/ conv. rate depending on regularity of exact solution
  - - - in space
      - abstract form
      - operator form
  - - - general (mass-matrix) \( M \) non-diagonal
      - vertex integration rules or non-conf. \( P_1\)-elements: \( M\) diagonal
      - reduce ODE to explicit form
  - - - explicit
        
        unstable when time step \( \tau>h^2 \)
        
        left-sided rectangle rule
      - implicit (time integration method)
        
        satisfies difference eq. (Le118)
        
        Convergence of time discretization method/Err of implicit method (Th120)
        
        right-sided rectangle rule
        
        A-stable
    - - A-stable (P134)
      - more accurate & higher convergence order \( O(\tau^2) \)
      - Runge-Kutta even better
- - - - Galerkin Isomorphism \( G:\mathbb{R}^N \rightarrow V_h : \underline{u} \rightarrow u = \sum u_i \varphi_i \)
        with dual \( G^*:V_h^* \rightarrow \mathbb{R}^N : d(\cdot ) \rightarrow (d( \varphi_i))_{i=1,...N} \)
      - eval. of quadratic form
        \( \underline{u}^T A \underline{u} = A(G\underline{u},G\underline{u}) \simeq \Vert G \underline{u} \Vert^2_{H^1} \)
      - Goal: construct \( C \) that
        
        precond. action \( w=C^{-1} \times d \) efficiently computable
        
        estimate spectral bounds
    - - preconditioning action \( \sum_{i=1}^N e_i ( e_i^T A e_i )^{-1}e_i^T \underline{d} \)
      - quadratic form \( \underline{u}^T C \underline{u} = \sum_{i=1}^N u_i^2 \Vert \varphi_i \Vert^2_A\)
      - min-mesh size \( h \) of a shape-regular triang.(Th79)
        \( \Rightarrow h^2 \underline{u}^T C \underline{u} \preceq u^T A u \preceq u^T C u \)
    - - goal:solve finite element system on \(V_L=V_h \subset H^1 \)
      - \( C=C_L\)
        
        recursively
        
        \( C_0^{-1}:= A_0^{-1}\)
        
        \( C_l^{-1}:=(\mathrm{diag}\,A_l)^{-1} + E_l C_{l-1}^{-1}E_l^T\)
        
        computation complexity of \( C_L^{-1}\) is \( O(N)\)
      - use finite element spaces \( V_0 \subset V_1 \subset ... \subset V_L \)
        on (hierarchy of) coarser meshes \( \mathcal{T}_0,\mathcal{T}_1,...,\mathcal{T}_L \)
      - simple analysis (Le1)
        \( \frac{1}{L}C \preceq A \preceq LC \)
      - improved estimate (Le2)
        =fulfills optimal spectral bounds
        
        leading to optimal condition number \( \kappa(C^{-1}A)\preceq 1\) independent of \( L \)
        
        intuitive explanation (P4)
        
        \( C \preceq A \preceq C \)
      - finite element spaces \( V_l \)
        
        \( h_l \simeq 2^{-l} \)
        
        \( N_l = \mathrm{dim}(V_l) \)
        
        \( \{ \varphi_{l,i}:1 \leq i \leq N_l \}\)hat basis
        
        \( A_l \) FE matrix
    - - rectangular embedding matrices \( E_i \in \mathbb{R}^{N \times N_i}, \, i=1,...,M \)
        \( \Rightarrow \) unique representation \( \underline{u}=\sum^M_{i=1} E_i \underline{u}_i \)
      - quadratic form \( \underline{u}^T C \underline{u} = \sum_{i=1}^M \Vert GE_i \underline{u}_i \Vert^2_A\)
      - Example on unit interval
      - preconditioning action \( \sum_{i=1}^M E_i ( E_i^T A E_i )^{-1} E_i^T \underline{d} \)
    - - precond. action same as in Block-Jacobi
      - Example on unit interval
      - Additive Schwarz Lemma = useful representation in quadr. form (Le80)
        \( \underline{u}^T C \underline{u} = \inf_{\underline{u}_i \in \mathbb{R}^{N_i}}\sum_{i=1}^M \underline{u}_i^T A_i \underline{u}_i\)
        with \( A_i = E_i^T A E_i \)
      - Add. Schwarz Lemma in finite element framework (= \( V_i, P_i, \widehat{u}\),...) (Le82)
        \( \underline{u}^T C \underline{u} = \inf_{u_i \in V_i} \sum_{i=1}^M \Vert u_i \Vert_A^2 \)
      - \( \Rightarrow \) (not necessarily) unique representation \( \underline{u}=\sum^M_{i=1} E_i \underline{u}_i \)
      - FE space by sub-space splitting
        \( V=\sum_{i=1}^M V_i, V_i=GE_i\mathbb{R}^{N_i}\subset V_h \)
    - - local (P89)
        
        ODD precond. fulfills spectral estimates (Le83)
        \( H^2 \underline{u}^T C \underline{u} \preceq \underline{u}^T A \underline{u} \preceq \underline{u}^T C \underline{u} \)
        
        Decomposition of \( \Omega=\cup_{I=1}^M \Omega_i\) with
        
        \( diam(\Omega_i)=H\)
        
        \( \Omega_i \subset \widetilde{\Omega}_i, \, \mathrm{dist}\{ \partial\widetilde{\Omega}_i \setminus \partial \Omega, \partial \Omega_i\} \succeq H \)
        
        only a small number of \( \widetilde{\Omega}_i \) overlap
        
        FE mesh with \( h \leq H \)
        
        FE space by sub-space splitting
        \( V_h=\sum V_i, V_i=V_h \cap H_0^1(\widetilde{\Omega}_i) \)
        
        in the limit , if \( H \simeq h\) it is comparable to Jacobi-precond.
      - ...w/ coarse grid correction (P90)
        
        problem: local ODD worse when number of sub-domains increase
        
        add one more sub spaces
        
        \( \mathcal{T}_H \) coarse mesh
        
        \( \mathcal{T}_h \) fine mesh by subdivision of \( \mathcal{T}_H \)
        
        \( V_H \) FE space on \( \mathcal{T}_H \)
        
        sub-space decomp. \( V_h=V_H + \sum _{i=1}^M V_i\)
        
        graphical Ex (P91)
        
        prolongation matrix \( E_H \)
        restriction matrix \( E_H^T \)
        
        preconditioning action \( \sum_{i=1} E_i ( E_i^T A E_i )^{-1} E_i^T \underline{d} + E_H ( E_H^T A E_H )^{-1} E_H^T \underline{d} \)
        
        it fulfills optimal spectral estimates (Th84)
        \( \underline{u}^T C \underline{u} \preceq \underline{u}^T A \underline{u} \preceq \underline{u}^T C \underline{u} \)
        
        optimal number of subdomains \( M=N^{\frac{\alpha}{2\alpha -1}} \)
        
        asymptotic cost \( N^{\frac{\alpha^2}{2\alpha -1}} \)+ Ex
  - - - 1D-Dirichlet on interval
      - 2D-Dirichlet on square
  - - - LU-decomposition
      - Cholesky factorization
      - band matrix -> faster
    - - block factorization & sub structuring algos
      - optimal number of sub-domains
      - result
        
        better than general
        
        2D very efficient
        
        3D worse
  - - - algo: \( u^{k+1}=u^k+\tau C^{-1}(f-Au^k) \)
        
        (optimal) damping \( \tau \)
        
        preconditioner matrix \( C \)
        
        iteration matrix \( M=I-\tau C^{-1}A \),
        with \( u^{k+1}-u = M(u^k - u) \)
      - \( M \) can be bound in \( A \)- and \( C \)-norm (Le75)
      - comparable to band factorization
    - - find automatically optimal \( \tau \) ,
        by minimum in convex function per step
      - convergence rate estimated by Richardson iteration
    - - optimization of \( \tau \) solved by chebyshev plynomials
      - original method not good choice
        
        not numerically stable
        
        know a priori \( N_{its} \)
      - \( m \) iterations \( \Rightarrow \) overall convergence better w/ variable \( \tau_1,...,\tau_m \)
    - - Algo
      - compute an \( A\)-orthogonal (=conjugate) basis of the Krylov space \( \mathcal{K}_k(d_0) \)
      - terminates at finite number of steps (Re76)
      - algorithm fulfills (Th77)
        
        seq. \( p_k \) is \( A\)-orth. and spans \( \mathcal{K}_k(d_0) \)
        
        \( u_k \) minimizes \( \min_{v \in u_0+\mathcal{K}_k(d_0)} \Vert u-v \Vert_A \)
        
        orth. \( d_k^T p_l = 0, \, \forall l< k \)
      - convergence rate (Th78)
      - error in energy norm
        
        goal: reduction by factor \( \epsilon \)
        
        can't be calculated \( \Rightarrow \) approx \( d^T_k C^{-1} d_k \)
- - - - norm
      - spaces \(V_s\)
      - \( V_1 \subset V_s \subset V_0 \)
  - - - norm
      - spaces \( V_s\)
      - K-functional
- - - - Lagrangian vs. Eulerian (Ch1)
        
        Eulerian
        
        \( x \) variables
        
        fluids
        
        Langrangian
        
        coordinates \( X \)
        
        solids
        
        variables
        
        velocity of particle \( u(x,t) \)
        
        position \( x \in \Omega \)
        
        particle \( X \in \Omega\)
        
        evolution of particles \( \varphi : \Omega\times [0,T] \rightarrow \Omega \)
        
        \( \Omega \in \mathbb{R} \)
        
        basis: conservation eq.s (e.g. mass)
        
        Reynolds transport Theorem (Th1)
        
        quantity preserved
        
        Interpretation
        
        material derivative
        \( \frac{Dc}{Dt} := \frac{\partial c}{\partial t} + \mathrm{div\,}(uc) \)
        
        \( \frac{d}{dt}\int_{V(t)} c(x,t)\mathrm{\,dx}= \int_{V(t)} \frac{\partial c}{\partial t}(x,t)\mathrm{\,dx}+ \int_{\partial V(t)} n\cdot uc\mathrm{\,ds} \)
      - The N-S.eq (Ch2)
        
        unknown vars
        
        vector valued velocity field \( u \)
        
        scalar-valued pressure field \( p \)
        
        conserved quants.
        
        mass
        
        momentum
        
        new var. form
      - 1st FEM discretization & time stepping (Ch3)
        
        space-discretized system
        
        implicit + explicit Euler method (mixture used)
        
        goal: rewrite terms to solve per time step
    - - stability of cont. eq (Ch1)
        
        stokes eq. = N-S-eq - viscos. & density terms
        
        solvability by Brezzi
        
        LBB-cond. non-trivial
        
        Proof1: ignore Dirichlet and use extension operator
        
        Proof2: on star shaped \( \Omega \)
      - Discrete LBB (Ch2)
        
        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
        
        \( V_h = P^{2}, \, Q_h = P^{0,dc} \)
        
        Fortin + Clement operator
        
        correction operator
        
        error-estimates
        
        1st order convergence
        
        this element sub-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
        
        Elements w/ cont. pressure (Ch2.2)
        
        mini element \( V_h = P^{1+}, \, Q_h = P^{1,cont} \)
        
        Fortin operator (preserving mean values)
        
        clement operator
        
        correction operator
        
        \( O(h) \) convergent
        
        Alternative: Taylor-Hood element
        
        stable pair \( P^2 \times P^{1,cont} \)
        
        requires add. assumptions
        
        conv. rate \( O(h^2) \)
    - - flow of a fluid
      - incompressible fluids
      - stationary = steady state (no change in \(t\))
    - - velocity \( u=(u_x, u_y, u_z) \)
      - pressure \( p\)
      - viscosity \( \nu \)
      - density \( \rho \)
      - model forces \( f\)
    - - var. problem: mixed form
      - w/ Brezzi and conditions \( \Rightarrow \) unique solution
      - inflow \( \Gamma_i \) and outflow boundaries \( \Gamma_o \)
    - - approx. by Galerkin
      - simplest elements not work (LBB not fulfilled)
      - spaces for convergence
        
        non-conf. \( P_1 \) for \( u \)
        
        piece-wise constant for \( p \)
      - proving discrete-LBB w/ Fortin operator
      - error of same order as approx. err
  - - - force density \( f \)
      - strain \( \epsilon = u'\)
      - stress \( \sigma' = -f\)
      - Hooks Law \( \sigma = E \epsilon \)
      - 2nd order eq. for displacement \( u \)
      - BC
    - - trafo of 1D setup
      - rigid body motions (=translation or linearized rotation)
      - eq. of elasticity in weak form
        \( u \in V= [H^1_{0,D}(\Omega)]^d\)
        \( \int_{\Omega} D\epsilon(u):\epsilon(v)\mathrm{\, dx}=\int_{\Omega}f\cdot v\mathrm{\, dx},\, \forall v \in V \)
      - BC
        
        Dirichlet (displacement at boundary)
        
        Neumann (stress at boundary )
      - weak form well posed in \( [H^1]^d \) (Th103)
      - discretization is straight forward
    - - problem: sim results on thin structures worse
      - Example sim: Timoshenko beam
      - semi discretizationgoal: derive system of 1D eq. for 2D deformation w/ semi-discretization
        
        average longitudinal displacing \( U(x) \)
        
        vertical displacement \( w(x) \)
        
        lin. rotation of normal vector \( \beta \)
        
        \( \Rightarrow \) 2 decoupled problems
        
        longitdunial displacement
        
        bending & shearing
        
        in principle well posed (Le104)
        
        small thickness
        \( \Rightarrow \) large cont./coerc. ratio
        \( \Rightarrow \) error increase
        
        idea: weaken term w/ large coefficient
        
        mixed method
        
        extended Brezzi (Th105)
  - - - electromagnetic fields
      - current density \( j \)
      - magnetic flux \( B\)
      - magnetic field intensity \( H \)
      - Henrys Law + Stokes
      - \( \mathrm{curl}=\mathrm{rot}=\nabla\times\)
      - permeability \( \mu \)
      - system of eq.s
      - compare to diffusion eq.
    - - finite elements in 3D
      - tetrahedral
      - cheaper Nedelec / edge-element
      - basis function \( \varphi_{E_i}\)
    - - \( H^{1}\overset{\nabla}{\longrightarrow}H(\textrm{curl})\overset{\textrm{curl}}{\longrightarrow}H(\textrm{div})\overset{\textrm{div}}{\longrightarrow}L^{2} \)
      - \( W_h\overset{\nabla}{\longrightarrow}V_h\overset{\textrm{curl}}{\longrightarrow}Q_h\overset{\textrm{div}}{\longrightarrow}S_h \) (Th110)
      - de'Rham complex diagram commutes (Th111)
- - - - single step methods
      - based on taylor expansion for \( u, \ddot{u}\)
      - ODE \( M\ddot{u}+Ku=f \)
      - solve lin. system w/ spd matrix
      - satsifies discrete energy conservation
      - equivalent stiffness matrix \( K_{eq} \)
        pos. def. \( \Rightarrow \) conservation proves stability
      - conditional (=max. time-step)/unconditional stability
    - - wave eq.
        
        reduce to 1st order system of PDEs
        
        mixed var. form.
        
        space discretization -> ODE system
      - conservation of energy
      - symplectic Euler method
      - skew symmetric matrix
        
        \( \Rightarrow \) EVa imaginary
        
        Hamiltonian structure
        
        symplectic methods
- - - - derivation (w/ boundary conditions) (P6)
      - functions & (their) spaces (P41)
        
        \( u_D \in H^{1/2}(\Gamma_D)\)
        
        \( f \in L_2 (\Omega)\)
        
        \( g \in L_2 (\Gamma_N \cup \Gamma_R )\)
        
        \( \alpha \in L_{\infty}(\Gamma_D), \alpha>0 \)
        
        HS \( V:=H^1(\Omega)\)
        
        closed subspace \( V_0=\left\{v:tr_{\Gamma_D}v=0\right\}\)
        
        linear manifold \( V_D=\left\{u \in V:tr_{\Gamma_D}u=u_D\right\}\)
      - Assumptions
        
        \( \left\vert \Gamma_D \right\vert > 0 \)
        
        \( \int_{\Gamma_R}\alpha \,\mathrm dx >0 \)
      - BLF \( A(\cdot,\cdot):V \times V \rightarrow \mathbb{R}\)
        \( A(u,v)= \int_\Omega \nabla u \nabla v \,\mathrm dx + \int_{\Gamma_R}\alpha u v \,\mathrm ds\)
      - LF \( f(v)= \int_\Omega fv \,\mathrm dx + \int_{\Gamma_N \cup \Gamma_R} gv \,\mathrm ds \)
      - weak. form \( A(u,v)=f(v), \, \forall v\in V_0\) has unique solution \( u \) (Th56) + Proof
    - - \( -\Delta u =f \) in \( \Omega \)
      - Boundary conditions
        \( u=u_D \) on \( \Gamma_D\)
        \( \frac{ \partial u}{ \partial n}=g \) on \( \Gamma_N\)
        \( \frac{ \partial u}{ \partial n} +\alpha u=g \) on \( \Gamma_R\)
      - \( \Omega \) bounded and open subset of \( \mathbb{R}^d \)
        \( \partial\Omega =: \Gamma = \Gamma_D +\Gamma_N + \Gamma_R \) non overlapping
      - \( \Delta:=\sum_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}} \)