[PPT] - Finite volume method for linear and non linear elliptic problems PowerPoint Presentation

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Finite volume method for linear and non linear elliptic problems with discontinuities

Franck BOYER et Florence HUBERT L.A.T.P . - Marseille, FRANCE DD17 - 2006

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SLIDE 2

Plan

PLAN

1 INTRODUCTION 2 THE DDFV SCHEME 3 THE 1D PROBLEM 4 THE 2D PROBLEM 5 A SADDLE-POINT ALGORITHM 6 NUMERICAL RESULTS

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SLIDE 3

Introduction

PLAN

1 INTRODUCTION 2

The DDFV scheme

3

The 1D problem

4

The 2D problem

5

A saddle-point algorithm

6

Numerical results

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SLIDE 4

Introduction

INTRODUCTION.

◮ DDFV scheme (DISCRETE DUALITY FINITE VOLUME) for

−div (ϕ(z, ∇ue(z))) = f(z),

in Ω, ue = 0, on ∂Ω, (1) Ω polygonal open set in R2. u → −div(ϕ(·, ∇u)) is coercitive, monotonous (of Leray-Lions type). ϕ presents discontinuities with respect to the space variable z (Transmission problem). Example :

Ω1 −div|∇u|p−2∇u = f1 −div “ k(x)|∇u|p−2∇u ” = f2 Ω2

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SLIDE 5

Introduction

INTRODUCTION

With discontinuities the DDFV scheme converges but slowly : In the linear case : 1D : −(λ±ue′)′ = f

Arythmetic mean-value : order 1

2

Harmonic mean-value : order 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 16 mailles, avec lambda=1si x<0.5 et 10 sinon exact moyenne arithmétique moyenne harmonique

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SLIDE 6

Introduction

INTRODUCTION

With discontinuities, the DDFV scheme converges but slowly : In the linear case : 2D : −div(A(z)∇ue) = f

DDFV scheme : order 1

2

Improved DDFV scheme (Hermeline, BH) : order 1

−3 10 −2 10 −1 10 −5 10 −4 10 −3 10 −2 10 −1 10

rdre 0.96
rdre 1.99

L∞ error

−3 10 −2 10 −1 10 −5 10 −4 10 −3 10 −2 10

rdre 1.04
rdre 2.02

L2 error

−3 10 −2 10 −1 10 −3 10 −2 10 −1 10

rdre 0.52
rdre 1

H1 error A = 10 2 2 1

.

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SLIDE 7

Introduction

BIBLIOGRAPHIE

Anisotropy problems with discontinuities

EGH (00), Hermeline (03)

Gradient reconstruction problems

“O-scheme”, “U-scheme” Aavatsmark, Lepotier,...... Gradient FV schemes. Eymard, Gallou¨ et, Herbin,... Mixed FV schemes. Droniou, Eymard (06) DDFV schemes. Coudi` ere (99), Hermeline (00), Domelevo & Omn` es (05), ABH (06), Pierre (06), Delcourte & al (06).......

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SLIDE 8

Introduction

HYPOTH`

ESES SUR ϕ

Let p ∈]1, ∞[, p′ =

p p−1 and f ∈ Lp′(Ω). ◮ p ≥ 2 in this talk.

ϕ : Ω × R2 → R2 is a Caratheodory function such that : (ϕ(z, ξ), ξ) ≥ Cϕ

|ξ|p − 1
,

(H1) |ϕ(z, ξ)| ≤ Cϕ

|ξ|p−1 + 1
.

(H2) (ϕ(z, ξ) − ϕ(z, η), ξ − η) ≥ 1 Cϕ |ξ − η|p. (H3) |ϕ(z, ξ) − ϕ(z, η)| ≤ Cϕ

1 + |ξ|p−2 + |η|p−2

|ξ − η|. (H4) ϕ is piecewisely lipschitz in z ⇒ Assumption (H5).

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SLIDE 9

The DDFV scheme

PLAN

1

Introduction

2 THE DDFV SCHEME 3

The 1D problem

4

The 2D problem

5

A saddle-point algorithm

6

Numerical results

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SLIDE 10

The DDFV scheme

THE DDFV SCHEME

The discrete unknowns : uT =

uM, uM∗

where uM = (uK)K∈M ,uM∗ = (uK∗)K∗∈M∗ The discrete gradient : ∇T uT is constant on diamond cells ∇T

DuT =

1 sin αD uL − uK |σ∗| ν + uL∗ − uK∗ |σ| ν∗

, ∀D.

uL+uK∗ 2

xL

uL+uL∗ 2 uK+uL∗ 2

xK

uK+uK∗ 2

xL∗ xK∗ 10/ 37

SLIDE 11

The DDFV scheme

THE DDFV MESHES

primal, dual and “diammond”.

xL∗ xK∗ K∗ L∗ K xL xK L maillage M maillage M∗ maillage D

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SLIDE 12

The DDFV scheme

THE DDFV SCHEME

−

Dσ,σ∗∩K=∅

|σ| (ϕD(∇T

DuT ), νK) =

K

f(z) dz, ∀K ∈ M, −

Dσ,σ∗∩K∗=∅

|σ∗| (ϕD(∇T

DuT ), νK∗) =

K∗ f(z) dz, ∀K∗ ∈ M∗,

with ϕD(ξ) = 1 |D|

D

ϕ(z, ξ) dz. Discrete Duality formulation : 2

D∈D

|D| (ϕD(∇T

DuT ), ∇T Dv T ) =

Ω

fvMdz +

Ω

fvM∗dz, ∀v T ∈ RT .

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SLIDE 13

The DDFV scheme

KNOWN RESULTS

Case p = 2 : Domelevo & Omn` es ⇒ Estimate in O(h) for a large class of meshes. General case : Andreianov, Boyer & Hubert ⇒ The scheme converges. ⇒ If ue ∈ W 2,p(Ω) and ϕ Lip. on Ω, with

∂ϕ

∂z (z, ξ)

≤ Cϕ
1 + |ξ|p−1

, ∀ξ ∈ R2. (H5) Then ue−uMLp+ue−uM∗Lp+∇ue−∇T uT Lp ≤ C size(T )

1 p−1 . 13/ 37

SLIDE 14

The DDFV scheme

ZOOM ON THE DIAMOND CELLS

Diamond cells are supposed to be convex. Each diamond cell is cut into four triangles Q.

|σL∗| |σK∗| xK |σK| αD ν∗ τ ∗ τ xL xL∗ |σL| ν xK∗ xK∗ QL,L∗ QL,K∗ xK xL QK,L∗ xD QK,K∗ xL∗

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SLIDE 15

The DDFV scheme

AIMS

If ϕ presents some discontinuities on a curve Γ ue ∈ W 2,p(Ω). The numerical fluxes are no more consistent along Γ . We assume that on each Q, ϕ is Lip. and saitifies (H5). ◮ We want to get the consistency of the fluxes on Γ. ◮ We contruct a new approximation ϕN

D of the non linearity on the

diamond cell D. −

Dσ,σ∗∩K=∅

|σ| (ϕN

D (∇T DuT ), νK) =

K

f(z) dz, ∀K ∈ M −

Dσ,σ∗∩K∗=∅

|σ∗| (ϕN

D (∇T DuT ), νK∗) =

K∗ f(z) dz, ∀K∗ ∈ M∗

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SLIDE 16

The 1D problem

PLAN

1

Introduction

2

The DDFV scheme

3 THE 1D PROBLEM 4

The 2D problem

5

A saddle-point algorithm

6

Numerical results

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SLIDE 17

The 1D problem

THE 1D PROBLEM

Ω =] − 1, 1[, ϕ(x, ·) = ϕ−(·), if x < 0, ϕ+(·), if x > 0 . Let x0 = −1 < . . . < xN = 0 < . . . < xN+M = 1 a discretization of [−1, 1]. The 1D FV scheme reads for i ∈ {0, N + M − 1} : − Fi+1 + Fi = xi+1

xi

f(x) dx. (2) with Fi = ϕ(xi, ∇iuT ), ∇iuT = ui+ 1

2 − ui− 1 2

xi+ 1

2 − xi− 1 2

, ∀i = N, (3)

QUESTION : how to define FN ?

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SLIDE 18

The 1D problem

THE NEW GRADIENT

We look for ˜ u such that ∇+

NuT =

uN+ 1

2 − ˜

u h+

N

, ∇−

N uT =

˜ u − uN− 1

2

h−

N

, we have ϕ−(∇+

NuT ) = ϕ+(∇− N uT ).

¯ u h+

N

h−

N

xN = 0 ˜ u δ uN− 1

2

uN+ 1

2

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SLIDE 19

The 1D problem

THE NEW GRADIENT

We look for ˜ u in the form ˜ u = ¯ u + δ, with ¯ u = h−

N uN+ 1 2 + h+ NuN− 1 2

h−

N + h+ N

. that is ∇+

NuT = ∇NuT − δ

h+

N

, and ∇−

N uT = ∇NuT + δ

h−

N

.

¯ u h+

N

h−

N

xN = 0 ˜ u δ uN− 1

2

uN+ 1

2

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SLIDE 20

The 1D problem

THE NEW GRADIENT

THEOREM For all uT ∈ RN, there exists a unique δN(∇NuT ) such that FN

def

= ϕ−

∇NuT + δN(∇NuT )

h−

N

= ϕ+
∇NuT − δN(∇NuT )

h+

N

,

The new scheme admits a unique solution. The flux FN is consistent with an error in h

1 p−1 . 19/ 37

SLIDE 21

The 1D problem

EXAMPLE

For two p−laplacian fluxes ϕ−(ξ) = k−|ξ + G−|p−2(ξ + G−), and ϕ+(ξ) = k+|ξ + G+|p−2(ξ + G+), where k−, k+ ∈ R+ and G−, G+ ∈ R2. We obtain FN =  k

1 p−1 −

k

1 p−1 +

(h−

N + h+ N)

h+

Nk 1 p−1 −

+ h−

N k 1 p−1 +

 

p−1

∇NuT + G
p−2

∇NuT + G

,

where G is some arythmetic mean value of G− and G+ defined by G = h−

N G− + h+ NG+

h−

N + h+ N

. Warning : the fluxes are not explicit in general !

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SLIDE 22

The 2D problem

PLAN

1

Introduction

2

The DDFV scheme

3

The 1D problem

4 THE 2D PROBLEM 5

A saddle-point algorithm

6

Numerical results

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SLIDE 23

The 2D problem

THE NEW GRADIENT

◮ ∇N

D uT is constant on each quarter of diamond

∇N

D uT =

Q∈QD

1Q∇N

QuT ,

xK xK∗ xσK xσK∗ QK,K∗ xD uσK uσK∗

1 2 (uK + uK∗)

∇

N QK,K∗ u T =

2 sin αD uσK∗ − 1

2(uK + uK∗)

|σK| ν + uσK − 1

2(uK + uK∗)

|σK∗| ν∗

That is ∇N

QuT = ∇T DuT + BQδD, δD ∈ R4,

where δD is to be determined BQK,K∗ = 1 |QK,K∗| (|σK|ν∗, 0, |σK∗|ν, 0)

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SLIDE 24

The 2D problem

CONSISTENCY OF THE FLUX

We note ϕQ(ξ) =

Q

ϕ(z, ξ) dµ¯

Q(z).

and choose δD ∈ R4 such that

ϕQK,K∗(∇T

DuT + BQK,K∗δD), ν∗

=

ϕQK,L∗(∇T

DuT + BQK,L∗δD), ν∗

ϕQL,K∗(∇T

DuT + BQL,K∗δD), ν∗

=

ϕQL,L∗(∇T

DuT + BQL,L∗δD), ν∗

ϕQK,K∗(∇T

DuT + BQK,K∗δD), ν

=
ϕQL,K∗(∇T

DuT + BQL,K∗δD), ν

ϕQK,L∗(∇T

DuT + BQK,L∗δD), ν

=
ϕQL,L∗(∇T

DuT + BQL,L∗δD), ν

⇒ For all uT ∈ RT , and all diamond cell D, there exists a unique

δD(∇T

DuT ) ∈ R4 that ensures such equalities. 23/ 37

SLIDE 25

The 2D problem

THE NEW SCHEME

ϕN

D (∇T DuT ) = 1

|D|

Q∈QD

|Q|ϕQ(∇T

DuT + BQδD(∇T DuT )),

(5) ϕQ(ξ) =

Q

ϕ(z, ξ) dµ¯

Q(z).

FV FORMULATION −

Dσ,σ∗∩K=∅

|σ| (ϕN

D (∇T DuT ), νK) =

K

f(z) dz, ∀K ∈ M −

Dσ,σ∗∩K∗=∅

|σ∗| (ϕN

D (∇T DuT ), νK∗) =

K∗ f(z) dz, ∀K∗ ∈ M∗

(6)

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SLIDE 26

The 2D problem

THE NEW SCHEME

ϕN

D (∇T DuT ) = 1

|D|

Q∈QD

|Q|ϕQ(∇T

DuT + BQδD(∇T DuT )),

(5) ϕQ(ξ) =

Q

ϕ(z, ξ) dµ¯

Q(z).

DISCRETE DUALITY FORMULATION : 2

D∈D

|D| (ϕN

D (∇T DuT ), ∇T Dv T ) = 2

Q∈Q

|Q| (ϕQ(∇N

QuT ), ∇N Qv T )

=

Ω

fvMdz +

Ω

fvM∗dz, ∀v T ∈ RT .

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SLIDE 27

The 2D problem

EXAMPLE

If ϕ is linear (ϕ(z, ξ) = A(z)ξ) and constant on each primal cells, we find Hermeline schemes for which the numerical fluxes can be explicited.

A(z) = λ(z)Id, λ constant on the primal cells, αD = π

2

(ϕN

D , ν)

= λKλL

|σK| |σK|+|σL|λK + |σL| |σK|+|σL|λL

uL − uK |σK| + |σL|, (ϕN

D , ν∗)

=

|σK∗|

|σK∗| + |σL∗|λK + |σL∗| |σK∗| + |σL∗|λL

uL∗ − uK∗

|σK∗| + |σL∗|.

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SLIDE 28

The 2D problem

MAIN RESULT

THEOREM We assume that ϕ satisfies (H5) on each diamond cell. The scheme (6), (5) admits a unique solution uT . More over if ue|Q ∈ W 2,p(Q), ∀Q, we have ue − uMLp + ue − uM∗Lp + ∇ue − ∇NuT Lp ≤ C size(T )

1 (p−1)2 .

If ϕ is dicontinuous along some curve Γ in Ω and if we use ϕN

D only

in a neighbourhood V of Γ and if ue|Q ∈ W 2,p(p−1)2(Q), ∀Q ⊂ V, we have ue − uMLp + ue − uM∗Lp + ∇ue − ∇NuT Lp ≤ C size(T )

1 (p−1) . 26/ 37

SLIDE 29

A saddle-point algorithm

PLAN

1

Introduction

2

The DDFV scheme

3

The 1D problem

4

The 2D problem

5 A SADDLE-POINT ALGORITHM 6

Numerical results

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SLIDE 30

A saddle-point algorithm

POTENTIAL CASE

If ϕ derives from a potentiel Φ ϕ(z, ξ) = ∇ξΦ(z, ξ), for all ξ ∈ R2 a.e. z ∈ Ω, Φ(z, 0) = 0, a.e. z ∈ Ω. PROPOSITION The solution uT of the scheme (6) is the unique minimum of JT (v T ) =

D∈D
Q∈QD

|Q|ΦQ(∇N

Qv T )−

K

|K|fKvK−

K∗

|K∗|fK∗vK∗, ∀v T ∈ RT (6) with ΦQ(·) =

Q Φ(z, ·)dµQ(z).

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SLIDE 31

A saddle-point algorithm

POTENTIAL CASE

PROPOSITION The couple (uT , (δD(∇T

DuT ))D) is the unique minimum of

JT ,∆(v T , ˜ δ) =

D∈D
Q∈QD

|Q|ΦQ(∇T

Dv T + BQ˜

δD) −

K

|K|fKvK −

K∗

|K∗|fK∗vK∗, ∀v T ∈ RT , ∀˜ δ ∈ ∆. (7)

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SLIDE 32

A saddle-point algorithm

SADDLE-POINT FORMULATION

We want to solve this non quadratic minimization problem with a saddle-point formulation (See Glowinsky & al.) LT ,∆

r

(v T , ˜ δ, g, λ) =

Q∈Q

|Q|ΦQ(gQ) +

Q∈Q

|Q|(λQ, gQ − ∇T

Dv T − BQ˜

δD) + r 2

Q∈Q

|Q|

gQ − ∇T

Dv T − BQ˜

δD

2

−

K

|K|fKvK −

K∗

|K∗|fK∗vK∗, ∀v T ∈ RT , ∀˜ δ ∈ ∆, ∀g, λ ∈ (R2)Q.

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SLIDE 33

A saddle-point algorithm

SADDLE-POINT FORMULATION

The lagragian LT ,∆

r

admits a unique saddle-point given by ϕQ(gQ) + λQ + r(gQ − ∇T

DuT − BQδD) = 0, ∀Q ∈ Q,

r

Q∈QD

|Q|tBQ(BQδD + ∇T

DuT − gQ) −

Q∈QD

|Q|tBQλQ = 0, ∀D ∈ D, gQ − ∇T

DuT − BQδD = 0, ∀Q ∈ Q,

r

Q∈Q

|Q|(∇T

DuT + BQδD − gQ, ∇T Dv T ) =

K

|K|fKvK +

K∗

|K∗|fK∗vK∗ +

Q∈Q

|Q|(λQ, ∇T

Dv T ), ∀v T ∈ RT . 31/ 37

SLIDE 34

A saddle-point algorithm

DECOMPOSITION-COORDINATION ALGORITHM

Let r > 0.

Step 1 : Find (uT ,n, δn

D) solution of

r

Q∈Q

|Q|(∇T

DuT ,n + BQδn

D − gn−1

Q

, ∇T

Dv T )

=

K

|K|fKvK +

K∗

|K∗|fK∗vK∗ +

Q∈Q

|Q|(λn−1

Q

, ∇T

Dv), ∀v T ∈ RT .

r

Q∈QD

|Q|tBQ(BQδn

D + ∇T

DuT ,n − gn−1 Q

) −

Q∈QD

|Q|tBQλn−1

Q

= 0, ∀D ∈ D.

Step 2 : On each Q, find gn

Q solution de

ϕQ(gn

Q) + λn−1 Q

+ r(gn

Q − ∇T DuT ,n − BQδn

D) = 0.

Step 3 : On each Q solve λn

Q define by

λn

Q = λn−1 Q

+ r(gn

Q − ∇T DuT ,n − BQδn

D).

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SLIDE 35

A saddle-point algorithm

CONVERGENCE OF THE ALGORITHM

Remark : it works even in the non potential case. THEOREM ∀r > 0, the algorithm converges towards the unique solution of the improved DDFV scheme. Example ⇒ Converges in 25 iterations.

10 20 30 40 50 60 70 80 90 100

−9

10

−6

10

−3

10 10

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SLIDE 36

Numerical results

PLAN

1

Introduction

2

The DDFV scheme

3

The 1D problem

4

The 2D problem

5

A saddle-point algorithm

6 NUMERICAL RESULTS

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SLIDE 37

Numerical results

EXEMPLE

Ω =]0, 1[×]0, 1[, triangular mesh, ue piecewise quadratic, p = 3 if z1 < 0.5, ϕ(z, ξ) = |ξ|p−2ξ, if z1 > 0.5, ϕ(z, ξ) = (Aξ, ξ)

p−2 2 Aξ, with A =

2 5

.
DDFV () and improved DDFV (♦)

−3 10 −2 10 −1 10 −5 10 −4 10 −3 10 −2 10

rdre 1.71
rdre 0.97

L∞ error

−3 10 −2 10 −1 10 −6 10 −5 10 −4 10 −3 10 −2 10

rdre 1.73
rdre 0.97

Lp error

−3 10 −2 10 −1 10 −3 10 −2 10 −1 10

rdre 1
rdre 0.31

W 1,p error

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SLIDE 38

Numerical results

EXAMPLE

Ω =]0, 1[×]0, 1[, triangular mesh, ue piecewise quadratic, p = 5 if z1 < 0.5, ϕ(z, ξ) = |ξ|p−2ξ, if z1 > 0.5, ϕ(z, ξ) = (Aξ, ξ)

p−2 2 Aξ, with A = 10 Id.

DDFV scheme () and improved DDFV scheme (♦)

−3 10 −2 10 −1 10 −3 10 −2 10 −1 10 10 1 10

rdre 1.75
rdre 0.98

L∞ error

−3 10 −2 10 −1 10 −3 10 −2 10 −1 10 10

rdre 1.72
rdre 0.98

Lp error

−3 10 −2 10 −1 10 −1 10 10 1 10 2 10

rdre 1.08
rdre 0.18

W 1,p error

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SLIDE 39

Numerical results

CONCLUDING REMARKS

Summary : The variational structure of the equation is preserved. The discontinuities are handled in a good way (order 1 for p = 2). The scheme can be solved by a efficient saddle-point algorithm. Remarks : It works in the same way if 1 < p ≤ 2. We can couple nonlinearity whose order depends on the

subdomain. For example : Darcy / Darcy-Fochheimer.

We plan to couple the iterative method with domain decomposition method.

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