Intro (Ch0)
Transport eq. (Ch1)
DG for 2nd order eq (Ch4)
Nitsche's Method (Ch3)
DG discretization (Ch2)
IIP(incomplete IP) - DG
denser stiffness matrix -> overcome by Hybrid DG methods
Advantages
NIP(non-symmetric interior penalty) - DG
requires more unknowns
(H)DG for stokes & N-S (Ch4.4)
Hybrid DG (HDG) (Ch4.1)
SIP(symmetric interior penalty) - DG (element-by-element)
solvability (Ch1.1)
Matching int. rules (Ch4.3)
instationary transport eq.
∂u∂t+div(bu)=f on Ω×(0,T)
Bassi Rebay (Ch4.2)
for interface conditions (Ch3.1)
combi of finite volume methods & FEM
DG BLF
inflow-boundary -> jump term
DG formulation
discrete inf-sup cond. holds (Th1) --> general solvability
1st order eq
\( \textrm{div}(bu)=f \) on \( \Omega \)
for Dirichlet BCs (Ch3.0)
replace with upwind-limit
DG finite element spaces + discrete norms
var. problem consistent to solution
not consistent for dual problem --> Aubin-Nitsche not applicable
BLF elliptic in any case \( \alpha > 0 \)
space \(V\) by (semi-)norm
\( \Vert u \Vert_V = \Vert b\nabla u \Vert_{L_2} \)
test space \( W=L_2 \)
derivation
considered as stationary transport in space-time
\( \textrm{div}_{x,t}(\widetilde{b}u)=f \) on \( \Omega \)
inflow-outflow isometrie
for \(f=0\) and \(v=u\)
Lehrenfeld-Trick
disadvantages of DG
add consistent. & stabilization(as before)
var. form.
conservation principle (outflow = production inside)
var problem
BLF & LF cont. + inf-sup cond. trivial
mean value of normal derivative & jump
convergence \( O(h) \)
cont. case disappears
var. form.
leads to pos.-def. matrices (gluing method)
alternative to MM (Mortar)
discont. case important
BLF elliptic on FE space (Le2)
Nitsche's Method (BLF, LF, norm)
source \( f\)
add consistent & stability terms (1st for symmetric BLF)
replace penalty term by BR norm
Dirichlet BC in weak sense and obtain pos.def. matrix
BCs (inflow /outflow)
err ana
wind \( b\)
disadvantage of IP-DG: Penalty term w/ \( \alpha \)
space-time problems -> no vortices
vortices in \( b \)
consistent terms for symmetric. & coerc.
stability argument
(order-edges)-1 with same order of convergence
add variables \( \widehat{u},\widehat{v} \) on inter-element facets
projector
higher # (non-zero terms per row in Matrix) --> HDG
natural norm
ensures conservation of \( L_2 \) norm in time
elliptic on \( V_h \)
higher # dofs
apply discrete stability & consistency
consistent to solution
not able to argue with cont. of \( A \) (because not cont.)
estimate interp. err in all terms of A