parabolic PDEs (Ch9)

problem (Ch9.0)

Space-time form. of parabolic eq.s (Supp)

semi-discretization (Ch9.1)

Time integration methods (Ch9.2)
?Method of lines = Time-stepping

initial-boundary value problem

$H^1((0,T),V;L_2)=\{ v \in L_2((0,T),V):v'\in L_2((0,T),V^*) \}$
$\Vert v \Vert^2_{H^1} = \Vert v \Vert^2_{X} + \Vert v' \Vert^2_{X^*}$
one dimensional Sobolev-space with range in a HS (Def113)

weak formulation

weak derivative (Def112)

unique solution (Th115)
depending cont. on $ f $ & $ u_0 $

function spaces $ X, X^*$ & norms (P130)

t-dependent vector

Galerkin discretization in space

Trace Theorem (Th114)
point eval. is cont.

Solvability of cont. problem (Ch1)

test functions

error-estimate + Ritz projector (Th116)

Discont. Galerkin method (Ch3)

trapezoid rule/method (explicit)

A first time-discretization method (Ch2)

Euler method

Err of integration rule estimation (Th119)

system of ODEs

basis + expand solution

in space

operator form

Family of uniformly cont. & elliptic operators
$ A(t):V \rightarrow V^* $

time integration method fulfills stability estimate (Le117)

goal: solving the system of ODEs

adding up generates new var. problem

give meaning to $u(T)$

is uniquely solvable (Th1=Lions)

mesh dependent norms

$ X_h=Y_h=P^{k,dc}(V) $

discrete problem is inf-sup stable on $ X_h\times Y_h $

BLF well defined & cont. on $ X_h\times Y_h $

plugin IC $ u_0 \Rightarrow IC $ part of var. form

spatial function space dim= $ \infty $

A-stable (P134)

general (mass-matrix) $ M $ non-diagonal

explicit

cannot prove discrete inf-sup cond.

discretize in time

reduce ODE to explicit form

Runge-Kutta even better

Integration by parts in time = time-slabs

parabolic eq. + var. form

abstract form

unstable when time step $ \tau>h^2 $

parabolic eq. for every time-slab

new variable for $u(T)$

both well posed

implicit Euler

more accurate & higher convergence order $ O(\tau^2) $

eq. for time-step
= trapezoidal method (Crank Nicolson)
A- but not L-stable

implicit (time integration method)

vertex integration rules or non-conf. $ P_1$-elements: $ M$ diagonal

discrete err w/ conv. rate depending on regularity of exact solution

restrict test-space

left-sided rectangle rule

Initial value is end of previous time-slab

first mesh $ \mathcal{T}$

2nd mesh-dependent form.

A- stable

L-stable

Convergence of time discretization method/Err of implicit method (Th120)

satisfies difference eq. (Le118)

A-stable

right-sided rectangle rule