finite volume methods for dissipative...

Finite volume methods

for dissipative problems

Claire Chainais-Hillairet

CEMRACS summer school

Marseille, July 2019, 15-20

General approach

Modeling

a PDE system

Approximation/simulation

a numerical scheme

convergence

Analysis of PDEsNumerical analysis

existence of a solution ? uniqueness ?

structural properties ?

positivity, dissipation of entropy, bounds,

asymptotic behaviors,...

General approach

Modeling

a PDE system


a numerical scheme

convergenceAnalysis of PDEs

Numerical analysis





General approach

Modeling

a PDE system


a numerical scheme

convergenceAnalysis of PDEsNumerical analysis





Outline of the course

1 Introduction to the finite volume method

2 Dissipative problems and long time behavior

3 Finite volume schemes and long time behavior

Lecture 1 :

Introduction to

the finite volume method

Outline of the chapter

1 Presentation of the finite volume methodBasic principlesSome examples

2 Analysis of the finite volume scheme for the Poisson equationSome remindersPresentation of the scheme and first propertiesConvergence of the scheme

For which type of equations ?

q Eymard, Gallouet, Herbin, 2000

ß Discretization in space of conservation laws

divJ = g.

ß For evolutive equations

∂tu+ divJ = g,

finite difference method for the time discretization.

Fluxes

Diffusion fluxes : J = ∇u, J = ∇r(u), J = K∇u,...

Convection fluxes : J = vu, J = vf(u), J = F(u),...

Every combination...

Mesh : definitions and notations

•

•

xL

xK

K

L

σ = K|L

nK,σ

Ω = (0, 1)× (0, 1)

T : set of control volumes K (open, convex, polygonal)

Ω =⋃K∈T

K, h = maxK∈T

diam(K)

E : set of edges σ

P : set of points xK ∈ K

Space of approximate solution

ß Reconstruction of piecewise constant approximate solutions

Discrete unknowns(uK)K∈T

Approximate solution

uT =∑K∈T

uK1K

Space of approximate solutions

X(T ) =

uT =

∑K∈T

uK1K

How to get a finite volume scheme ?

•

•

xL

xK

K

L

σ = K|L

nK,σ

Integration over each cell K of

divJ = g

⇓∑σ∈EK

∫σJ · nK,σ =

∫Kg

Finite volume scheme∑σ∈EK

FK,σ = m(K)gK for all K ∈ T ,

with

FK,σ “good” approximation of

∫σJ · nK,σ,

gK =1

m(K)

∫Kg (or an approximation).

Crucial properties of the numerical fluxes

K

Lσ = K|L

nK,σ

Conservativity of the numerical fluxes

FK,σ + FL,σ = 0 ∀σ = K|L.

Consistency of the numerical fluxes

FK,σ : evaluation of the numerical flux for an exact andsmooth solution of the problem

J : exact flux

1

m(σ)

(FK,σ −

∫σJ · nK,σ

)= O(h).

Diffusion flux : J = −∇u

•

•

xL

xK

K

L

σ

dσ nK,σ

dσ = d(xK , xL)

J = −∇u

⇓∫σJ · nK,σ = −

∫σ∇u · nK,σ

FK,σ = −m(σ)uL − uKd(xK , xL)

= −m(σ)

dσ(uL − uK).

Conservativity : OK

Consistency : OK... if (xKxL) ⊥ σ

ß Admissible mesh : ∀σ = K|L, (xKxL) ⊥ σ

Convection flux : J = vu

•

•

xL

xK

K

L

σ

nK,σ

J = vu

⇓∫σJ · nK,σ =

∫σv · nK,σu

vK,σ =1

m(σ)

∫σv · nK,σ

Centered fluxes

FK,σ = m(σ)vK,σuK + uL

2Upwind fluxes

FK,σ = m(σ)vK,σ

uK if vK,σ ≥ 0uL if vK,σ ≤ 0

Convection flux : J = vu

Centered fluxes

FK,σ = m(σ)vK,σuK + uL

2

Upwind fluxes

FK,σ = m(σ)vK,σ

uK if vK,σ ≥ 0uL if vK,σ ≤ 0

Conservativity OK (vK,σ + vL,σ = 0)

Consistency OK

ß Without any assumption on the mesh.

ß But : well-known instability problems with the centeredscheme for pure convection equations.

ß A monotony hypothesis is needed in this case.

Theoretical reminder

Problem under study

Ω regular bounded open subset of Rd, f ∈ L2(Ω)−∆u = f in Ω,

u = 0 on Γ = ∂Ω.

Theorem

There exists a unique u ∈ H10 (Ω) such that

∀v ∈ H10 (Ω)

∫Ω∇u · ∇v =

∫Ωfv.

Proof : application of Lax-Milgram lemma.

Theoretical reminder

Key points

ß Poincare inequality : ∃CP (Ω) such that ∀u ∈ H10 (Ω)∫

Ω|u|2 ≤ CP (Ω)

∫Ω|∇u|2.

ß |u|H10 (Ω) =

(∫Ω|∇u|2

)1/2

is a norm on H10 (Ω).

Qualitative properties

Monotony : v = u− in the weak formulation

f ≥ 0 a.e. =⇒ u ≥ 0 a.e.

Energy estimate : v = u in the weak formulation∫Ω|∇u|2 ≤ CP (Ω)

∫Ωf2

Finite element scheme / finite volume scheme

Conformal finite element scheme

Approximation of H10 (Ω) by finite-dimension subspaces :

(Vn)n≥0 such that Vn ⊂ H10 (Ω) ∀n

The scheme :

∀vn ∈ Vn∫

Ω∇un · ∇vn =

∫Ωfvn.

Poincare inequality can be applied.

Compact embedding of H1(Ω) into L2(Ω).

Finite volume scheme

X(T ) 6⊂ H1(Ω)

Discrete counterpart of Poincare inequality ?

Compactness arguments must be adapted.

The scheme

•

•

xL

xK

K

L

σ = K|L

dσ nK,σ

−∆u = f in Ω,

u = 0 on Γ = ∂Ω.

FK,σ ≈ −∫σ∇u · nK,σ

∑σ∈EK

FK,σ = m(K)fK ∀K ∈ T ,

FK,σ =

− m(σ)

dσ(uL − uK) if σ ∈ Eint, σ = K|L,

− m(σ)

dσ(0− uK) if σ ∈ Eext.

(dσ = d(xK , σ) if σ ∈ Eext)

Matricial form of the scheme∑σ∈EK

FK,σ = m(K)fK ∀K ∈ T

FK,σ = −τσ(uK,σ − uK) ∀K ∈ T ,∀σ ∈ EK .

with τσ =m(σ)

dσand uK,σ =

uL if σ ∈ EK,int, σ = K|L,0 if σ ∈ EK,ext.

Linear system of equations AU = B

Unknown : U = (UK)K∈T ,

Right hand side : B = (BK)K∈T with BK = m(K)fK ,

Matrix : A = (AK,L)K∈T ,L∈T

AK,K =∑σ∈EK

τσ

AK,L = −τσ if ∃σ = K|L, 0 else

Properties of the matrix AProposition 1 : A is a positive definite symmetric matrix

ß Calculation of UTAU for U ∈ R|T |

UTAU =∑K∈T

uK(AU)K

=∑K∈T

uK∑σ∈EK

FK,σ

=∑K∈T

uK∑σ∈EK

(−τσ(uK,σ − uK))

=∑

σ∈Eint,σ=K|L

τσ(uL − uK)2 +∑K∈T

∑σ∈EK,ext

τσu2K

ß UTAU ≥ 0 ∀U ∈ R|T | and UTAU = 0 =⇒ U = 0.

Þ Existence and uniqueness of a solution.

Properties of the matrix AProposition 2 : A is monotone : B ≥ 0 =⇒ U ≥ 0

ß Calculation of (U−)TAU for U ∈ R|T | (s− = min(s, 0))

(U−)TAU =∑K∈T

u−K(AU)K

=∑K∈T

u−K

∑σ∈EK

FK,σ

=∑K∈T

u−K

∑σ∈EK

(−τσ(uK,σ − uK))

=∑

σ∈Eint,σ=K|L

τσ(u−L − u−K)(uL − uK)

+∑K∈T

∑σ∈EK,ext

τσ(u−K)2

ß B ≥ 0 =⇒ (U−)TB = (U−)TAU ≤ 0 =⇒ U− = 0 and U ≥ 0.

Discrete energy estimate ?

AU = B =⇒ UTAU = UTB

∑σ∈Eint,σ=K|L


∑σ∈EK,ext

τσu2K =

∑K∈T

m(K)fKuK

≤

(∑K∈T

m(K)f2K

)1/2(∑K∈T

m(K)u2K

)1/2

Discrete Poincare inequality ? (admitted yet)

∑K∈T

m(K)u2K ≤ CdP

∑σ∈Eint,σ=K|L


∑σ∈EK,ext

τσu2K

Discrete energy estimate ?

(uK)K∈T solution to the scheme satisfies : ∑σ∈Eint,σ=K|L


∑σ∈EK,ext

τσu2K

1/2

≤ CdP ‖f‖L2(Ω).

Norms on X(T ) uT =∑K∈T

uK1K

‖uT ‖2L2(Ω) =∑K∈T

m(K)u2K

|uT |21,2,Γ,T =∑

σ∈Eint,σ=K|L


∑σ∈EK,ext

τσu2K

Þ Discrete H1 estimate for the approximate solution.

Construction of a sequence of approximate solutions

Sequence of meshes (Tn, En,Pn)n≥0

Admissible meshes,

limn→∞

hn = 0.

Sequence of approximate solutions (un)n≥0

un = uTn ∈ X(Tn).

Discrete H1 estimate

There exists C, not depending on n, such that

|un|1,2,Γ,Tn ≤ C ∀n ≥ 0.

Main result

Theorem

Let

(Tn, En,Pn)n≥0 a sequence of admissible meshes, withlimhn = 0,

(un)n≥0 a sequence of approximate solutions given by thefinite volume scheme.

Then,limn→∞

un = u in L2(Ω)

with

u ∈ H10 (Ω),

and u is the unique solution to the Poisson equation :

∀v ∈ H10 (Ω)

∫Ω∇u · ∇v =

∫Ωfv.

Proof of the theorem : the main steps

ß Compactness :

Consequence of the discrete H1 estimate,

Application of Riesz-Frechet-Kolmogorov theorem.

ß Strong convergence in L2(Ω) of a subsequence.

ß Let u be the limit : u ∈ H10 (Ω).

ß Passage to the limit in the numerical scheme to show that u isa weak solution.

ß Uniqueness of the solution =⇒ convergence of the wholesequence.

About the compactness step

Compactness theorem by Kolmogorov

Let Ω a bounded domain of RN , N ≥ 1.

Let A ⊂ Lq(Ω), 1 ≤ q < +∞.

A is relatively compact in Lq(Ω) if and only if thereexists p(u), u ∈ A ⊂ Lq(RN ) such that :

1 p(u) = u a.e. in Ω, for all u ∈ A,

2 p(u), u ∈ A is bounded in Lq(RN ),

3 ‖p(u)(·+ η)− p(u)(·)‖Lq(RN ) −→ 0 when η tends to 0,uniformly with respect to u ∈ A.

Application to the scheme

A = uT ; solution to the scheme on (T , E ,P) ⊂ L2(Ω)

p(uT ) =

uT on Ω0 on RN \ Ω

1 satisfied by definition

2 p(uT ), uT ∈ A is bounded in L2(RN ) :

‖p(uT )‖L2(RN ) = ‖uT ‖L2(Ω)

3 It can be proved that

‖p(uT )(·+ η)− p(uT )(·)‖2L2(RN ) ≤ |η|(|η|+ 2h)|uT |21,2,Γ,T .

q Eymard, Gallouet, Herbin, 2000

q Gallouet, Latche, 2013

finite volume methods for dissipative...

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