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 />
Convergence proof...<br />
The above points can be combined into a convergence proof:<br />
weak-* L ∞ compactness of (u T,∆t )<br />
=⇒ u T,∆t (·) “converges” to 1<br />
µ(·;α) dα,<br />
0<br />
the numerical convection flux converges to 1<br />
0 F(µ(·;α)) dα<br />
strong L 1 compactness of (w T,∆t ), weak L p compactness of (∇ T w T,∆t )<br />
=⇒ A(u T,∆t ) tends to w strongly, ∇ T A(u T,∆t ) tends to ∇w weakly,<br />
<strong>and</strong>a0(∇ T A(u T,∆t )) converges to some χ weakly.<br />
In addition, A(µ(·;α)) = w(·) for all α.<br />
And we have discrete entropy inequalities (with vanishing remainder<br />
terms) <strong>and</strong> discrete weak formulation where we can pass to the limit,<br />
using the above convergences + consistency of ∇ T on test functions .<br />
From the weak formulation we can identifyχ toa0(∇w) using the<br />
Minty-Browder argument. As a byproduct, we get strong L p convergence<br />
of ∇ T w T,∆t to ∇w.<br />
From the entropy formulation we conclude that u is an entropy-process<br />
solution , <strong>and</strong> use the reduction theorem to see that µ(·;α) ≡ u(·).