abstract theory (Ch2)

BASICS (Ch2.1)

Projection onto Subspaces (Ch2.2)

Riesz Representation Theorem (Ch2.3 & Th25)

Symmetric variational problems (Ch2.4)

Inf-Sup stable variations (Ch2.6)

Coercive variational Problems (Ch2.5)

Vector Space $V$ (Def1)

projection $ P^2=P $ (Def23)

Examples

Cauchy Schwarz inequality (Le9)

Examples

Approximation of inf-sup stable var problems (Ch2.6.1)

unique closest point $ u_0 $ in closed subspace $ S $ of HS $ V $ (Th21)

BLF symmetric

weaker condition to ensure $ B $ is onto $ W^* $ (=surjective) (eq2.11)

Approximation by Galerkin (Ch2.5.1)

inf-sup condition = generalization of coercivity (eq2.8)

finite element subspace $ V_h \subset V $

$ C^1(\Omega) $

form

Banach contraction mapping theorem (Th29)

more general BLF
$ B(\cdot,\cdot):V\times W \rightarrow \mathbb{R}$
(e.g. non-coercive)(P21)

BLF $B$ fullfills inf-sup condition

Lax Milgram (Th30)

normed (Def2)

functional = linear form (LF) (Def5)

linear operator(Def15)

dense subspace (Def17)

bilinear form (BLF)(Def6)

Hilbert space (HS)(Def11)

orthogonal $ \left(Pu,v \right) = \left(u,Pv \right) $ (Def23)

Diffusion-reaction eq. (Ex26)

coercive BLF (Ex34)

with Riesz $ \Rightarrow \exists!u:A\left(u,v\right)=f\left(v\right)\Rightarrow $ weak form has unique solution

complex symmetric var. problem (Ex35)

quasi optimal error estimate (Th36)

$ ( C^1(\Omega), \left\Vert \cdot \right\Vert _{A})$ not complete

Diffusion-convection-reaction eq. (Ex27)

$ V=\overline{C^{1}(\Omega)}^{\left\Vert \cdot\right\Vert _{A}} $ is HS

Cea - error is quasi optimal (Th31)

solvability of finite dim eq does not follow from solvability of original problem --> extra inf-sup condition for discrete problem

error $ u-u_h $ is orth. to $ V_h $

& weaker condition is fulfilled $ \Rightarrow $ problem has unique solution, which depends cont. on right side (Th33)

$ u_h $ is projection of $ u $ on $ V_h $

$ \Rightarrow $ operator B has closed range (Le32)

if bounded $ \Rightarrow $ cont.

$V$ is HS

unique solvability (inherited from original problem)

$ dim(V_h)<\infty \Rightarrow closed \Rightarrow$ use Th21

non-neg

$ A(u,u)=0 \Rightarrow u=0$

LF cont

closure $ \overline{W}^{\left\Vert \cdot\right\Vert _{V}} $ (Def4)

$ \Rightarrow $ unique decomposition of $ u \in V $
$ u=u_0 + u_1 $
$ u_0\in S , u_1 \in S^\perp $

$ \Rightarrow B$ is one-to-one (=injective)

BLF coercive aka elliptic & cont.

dual norm (Def5)

linear operator $B:V \rightarrow W^* $ by $ \left\langle Bu,v \right\rangle_{W^*\times W}=B(u,v) $ as in Le28 (P22)

if BLF A is symmetric $ \Rightarrow $ factor in error-estimate is sqrt of Th31

closed subspace (Def12)

Notation: duality product $ \left\langle f,v\right\rangle_{V^* \times V} =f\left(v\right) $ (P18)

complete = Banach Space (BS) (Def3)

operator norm(Def15)

$ \Rightarrow $ define operators $ P_S, P_S^\perp $

rewrite var problem as operator eq. (P19)
Find $u$ that $ Au=f $ in $ V^*$

$ \Rightarrow A(\cdot,\cdot) $ is IP and $ \left\Vert \cdot\right\Vert _{A} $ is norm

bounded (Def5)

bounded (Def15)

symmetric

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

linear (Th22)

dual space $ V^* $ (Def5)

$ \Rightarrow $ continous (Le16)

orthogonal (Th24)

compact (Def19)

$ \ker T:=\left\{ v\in V:Tv=0 \right\} $ is closed subspace of HS V, where T is cont. lin. operator (Le13)

extension principle for operators (Le18)

inner product $ (\cdot , \cdot)_V $ (IP) (Def7)

operator compact $ \Leftrightarrow \exists $ Eigensystem of the operator (Le20)

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) $