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abstract theory (Ch2) (Inf-Sup stable variations (Ch2.6) (Examples…
abstract theory (Ch2)
BASICS (Ch2.1)
Vector Space \( V \) (Def1)
normed (Def2)
closure \( \overline{W}^{\left\Vert \cdot\right\Vert _{V}} \) (Def4)
complete = Banach Space (BS) (Def3)
functional = linear form (LF) (Def5)
dual norm (Def5)
Notation: duality product \( \left\langle f,v\right\rangle_{V^* \times V} =f\left(v\right) \) (P18)
bounded (Def5)
dual space \( V^* \) (Def5)
linear operator(Def15)
operator norm(Def15)
bounded (Def15)
\( \Rightarrow \) continous (Le16)
compact (Def19)
operator compact \( \Leftrightarrow \exists \) Eigensystem of the operator (Le20)
extension principle for operators (Le18)
dense subspace (Def17)
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)
Cauchy Schwarz inequality (Le9)
Projection onto Subspaces (Ch2.2)
projection \( P^2=P \) (Def23)
orthogonal \( \left(Pu,v \right) = \left(u,Pv \right) \) (Def23)
unique closest point \( u_0 \) in closed subspace \( S \) of HS \( V \) (Th21)
\( \Rightarrow \) 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)
Riesz Representation Theorem (Ch2.3 & Th25)
Symmetric variational problems (Ch2.4)
LF
with Riesz \( \Rightarrow \exists!u:A\left(u,v\right)=f\left(v\right)\Rightarrow \) weak form has unique solution
if bounded \( \Rightarrow \) cont.
BLF
symmetric
non-neg
\( A(u,u)=0 \Rightarrow u=0\)
\( \Rightarrow A(\cdot,\cdot) \) is IP and \( \left\Vert \cdot\right\Vert _{A} \) is norm
finite element subspace \( V_h \subset V \)
error \( u-u_h \) is orth. to \( V_h \)
\( u_h \) is projection of \( u \) on \( V_h \)
\( dim(V_h)<\infty \Rightarrow closed \Rightarrow\) use Th21
\( C^1(\Omega) \)
\( ( C^1(\Omega), \left\Vert \cdot \right\Vert _{A})\) not complete
\( V=\overline{C^{1}(\Omega)}^{\left\Vert \cdot\right\Vert _{A}} \) is HS
Inf-Sup stable variations (Ch2.6)
Examples
coercive BLF (Ex34)
complex symmetric var. problem (Ex35)
Approximation of inf-sup stable var problems (Ch2.6.1)
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
weaker condition to ensure \( B \) is onto \( W^* \) (=surjective) (eq2.11)
inf-sup condition = generalization of coercivity (eq2.8)
\( \Rightarrow B\) is one-to-one (=injective)
more general BLF
\( B(\cdot,\cdot):V\times W \rightarrow \mathbb{R}\)
(e.g. non-coercive)(P21)
linear operator \(B:V \rightarrow W^* \) by \( \left\langle Bu,v \right\rangle_{W^*\times W}=B(u,v) \) as in Le28 (P22)
BLF \(B\) fullfills inf-sup condition
& weaker condition is fulfilled \( \Rightarrow \) problem has unique solution, which depends cont. on right side (Th33)
\( \Rightarrow \) operator B has closed range (Le32)
Coercive variational Problems (Ch2.5)
Examples
Diffusion-reaction eq. (Ex26)
Diffusion-convection-reaction eq. (Ex27)
Approximation by Galerkin (Ch2.5.1)
Cea - error is quasi optimal (Th31)
unique solvability (inherited from original problem)
if BLF A is symmetric \( \Rightarrow \) factor in error-estimate is sqrt of Th31
form
\(V\) is HS
LF cont
BLF coercive aka elliptic & cont.
rewrite var problem as operator eq. (P19)
Find \(u\) that \( Au=f \) in \( V^*\)
Banach contraction mapping theorem (Th29)
Lax Milgram (Th30)