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

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Degenerate Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators and scheme Discrete calculus & Convergence analysis Discrete calculus tools... Let’s follow the steps of the “continuous” convergence proof, looking at the discrete analogues of the arguments. “Variational” arguments: take w T for test function, get Energy estimates (=⇒ Existence + Weak L p compactness for gradients ∇ T w T,∆t + Estimate of space translates for w T,∆t ) Estimate of time translates for w T,∆t . We have to establish that (divc F) T (·) “coexists nicely” with variational technique , i.e., (div c F) T (u T ), A(u T ) behaves more or less like = Ω div F(u) A(u) := − F(u)· ∇A(u) Ω div u F(s) dA(s) = u F(s) dA(s) ·ν = 0. Ω 0 We also have to produce discrete versions of L p (0, T; W 1,p (Ω)) weak compactness, of Sobolev embeddings of W 1,p (Ω) into L sthg (Ω) (Andr. & Boyer & Hubert ), and exploit discrete duality. ∂Ω 0
Degenerate Parabolic Problems & FV Discretization Theoretical foundations Meshes, operators and scheme Discrete calculus & Convergence analysis Discrete calculus tools... “Entropy” arguments: take (η ± c ) ′ (u T ) for test function, get Discrete entropy inequalities L ∞ bound (from comparison with constant solutions) We already know that the discrete convection operator with monotone flux leads to discrete entropy inequalities with remainder terms controlled by the “weak BV” estimate , Eymard, Gallouët, Herbin . In addition, we have to establish that div T a0(∇ T ·) “coexists nicely” with the entropy technique , i.e., − div T k(∇ T A(u T ))∇ T A(u T ), θ(u T )ψ T (with θ = (η ± c ) ′ , ψT 0) behaves more or less like − div k(∇A(u))∇A(u)·(θ(u)ψ) := k(∇A(u))∇A(u)· ∇ Ω Ω θ(u)ψ k(∇A(u)) θ(u)∇A(u) · ∇ψ = k(∇A(u))∇ Aθ(A(u))· ∇ψ Ω Here we need to replace the chain rule by a convexity inequality and assume the orthogonality of the meshes. Ω
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Degenerate nonlinear parabolic-hyperbolic equations and ... - SMAI

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