Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI
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<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 discrete gradient operator is of the form<br />
∇ T : R T<br />
0 −→ (R d ) D ,<br />
where the values ∇Dw T are reconstructed<br />
per diamond from two projections: in R 2 ,<br />
∇Dw T =<br />
wL − wK<br />
dKL<br />
νK,L +<br />
wL ∗ − wK ∗<br />
dK ∗ L ∗<br />
νK ∗ ,L<br />
∗ for D = DK,L<br />
NB: The 3D case offers several choices:<br />
different constructions due to<br />
Pierre & Coudière ; Coudière & Hubert ;<br />
Hermeline, Andr. & Bendahmane & Hubert & Krell .<br />
We follow the last one, called “3D CeVe-DDFV”. primal<br />
The DDFV schemes enjoy discrete duality:<br />
Proposition (discrete duality)<br />
For v T ∈ R T<br />
0 <strong>and</strong> F T ∈ (R d ) D ,<br />
dual<br />
interface<br />
K ∗ |L ∗<br />
Diamond D K,L<br />
x L<br />
x K ∗<br />
x L ∗<br />
x K<br />
interface<br />
K|L<br />
<br />
− div T [ F T ], v T<br />
<br />
= F T T T<br />
, ∇ v .<br />
With this property, all “variational” techniques can be used at the discrete<br />
level ! The discretization of the Leray-Lions diffusion operator− diva0(∇·) by<br />
− div T a0(∇ T ·) preserves the key features of the continuous operator:<br />
coercivity, monotonicity, growth, existence of a potential...