Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI

Degenerate Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators and scheme Discrete calculus & Convergence analysis
Finite volume meshes and operators...
The space of discrete functions wT =
(wK) K ; (wK∗) K∗ is denoted
by R T , for functions zero on the boundary we use R T
0 .
The set of discrete fields ( F D) D is denoted (R d ) D .
On spaces R T and R D , we introduce scalar products
w T , v T
= 1
d
mK wKvK +
K∈M
and
F T
, G T
d − 1
d
K ∗ ∈M ∗
=
m DFD · GD; D∈D
m K ∗ w K ∗v K ∗
The discrete divergence operator is the usual Finite Volumes’ one:
we apply the Green-Gauss formula in each primal cell K and in each
dual cell K ∗ :
div T : (R d ) D −→ R T , with e.g. (div T
) KF := F D ·ν K.
D∈D
∂K∩D