Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Degenerate</strong> Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators <strong>and</strong> scheme Discrete calculus & Convergence analysis<br />
Finite volume meshes <strong>and</strong> operators...<br />
The space of discrete functions wT = <br />
(wK) K ; (wK∗) K∗ is denoted<br />
by R T , for functions zero on the boundary we use R T<br />
0 .<br />
The set of discrete fields ( F D) D is denoted (R d ) D .<br />
On spaces R T <strong>and</strong> R D , we introduce scalar products<br />
<br />
w T , v T<br />
<br />
= 1<br />
d<br />
<br />
mK wKvK +<br />
K∈M<br />
<strong>and</strong> <br />
F T<br />
, G T<br />
d − 1<br />
d<br />
<br />
K ∗ ∈M ∗<br />
= <br />
m DFD · GD; D∈D<br />
m K ∗ w K ∗v K ∗<br />
The discrete divergence operator is the usual Finite Volumes’ one:<br />
we apply the Green-Gauss formula in each primal cell K <strong>and</strong> in each<br />
dual cell K ∗ :<br />
div T : (R d ) D −→ R T , with e.g. (div T <br />
<br />
) KF := F D ·ν K.<br />
D∈D<br />
∂K∩D