2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Formulation and analysis of Discrete
Duality Finite Volume schemes.
Part II. DDFV schemes in 2D and 3D.
B. Andreianov1
based on joint works with
Franck Boyer and Florence Hubert (Marseille), Stella Krell (Nice),
Mostafa Bendahmane (Bordeaux), Kenneth H. Karlsen (Oslo).
Original ideas: F. Hermeline – K. Domelevo and P. Omnès (2D)
Ch.Pierre – F. Hermeline – Y. Coudière and F. Hubert (3D)
J. Droniou – R. Eymard – R. Herbin (DDFV ⊂ gradient schemes)
1
Université de Franche-Comté, Besançon, France
Rocquencourt, July 2015
CEA-EDF-INRIA School New Trends in Compatible Discretizations
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Plan of the talk
1
2D diamond partition and DDFV discrete gradient
Quadrilateral diamonds and a gradient reconstruction formula
Centers of diamonds, elements and DDFV volumes
2
2D DDFV scheme
Scalar products and Discrete Duality
Some properties of DDFV operators and schemes
3
The 3D CeVe-DDFV scheme
A primal mesh-oriented construction
3D CeVe-DDFV: scalar products, discrete duality and
reconstruction.
4
The 3D CeVeFE-DDFV scheme
5
Successful applications of DDFV schemes
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
2D DIAMOND PARTITION
AND DISCRETE GRADIENT
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Quadrilateral diamonds and a gradient reconstruction formula
Partition into quadrilateral diamonds
Partition Ω into quadrilateral (possibly degenerate) diamonds.
Quadrilateral has 4 vertices
= 2 pairs of “opposite” vertices.
Call them K , L and K ∗ , L∗ .
σ=
x K∗
ν∗
τ∗
τ
αD
xL∗
σ∗ =
K∗ |L∗
σ = K|L
σ∗ =
xK
ν
xK
xK∗
K|L
xL
xL
xL∗
K∗ |L∗
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Quadrilateral diamonds and a gradient reconstruction formula
2D DDFV gradient reconstruction
σ=
x K∗
ν∗
τ∗
τ
αD
σ∗ =
σ = K|L
σ∗ =
xK
ν
xK
xK∗
K|L
K∗ |L∗
K∗ |L∗
xL
xL
xL∗
xL∗
Attach DOFs uK , uL and u K ∗ , uL∗ to the couples of opposite vertices.
~ T :=
DDFV gradient reconstruction: ∇u
1
sin αD
Lemma (consistency of the 2D DDFV gradient)
This reconstruction is exact on affine functions.
Proof: Take scalar products by ~τ , ~τ ∗ .
uL − uK
uL∗ − uK∗ ∗
ν+
ν
mσ ∗
mσ
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Centers of diamonds, elements and DDFV volumes
Centers of diamonds, elements and DDFV volumes.
In order to create a FV scheme, one should attach volumes to DOFs.
Specificity of DDFV: overlapping volumes.
Two partitions: M (“primal” volumes K ) and M∗ (“dual” volumes K ∗ ).
Given a choice of xD “centers” of diamonds D,
the volumes are constructed from “elements”.
2D DDFV Element:
triangle with vertices xD , one ∈ {xK , xL }, and one ∈ {xK ∗ , xL∗ }.
“Primal” and “dual” 2D DDFV volumes:
Assemble K from all elements having K for vertice.
Idem for K ∗ = union of elements having K ∗ for vertice.
“Primal” mesh M := {all K }; “dual” mesh M∗ := {all K ∗ }.
Boundary primal and dual volumes (possibly degenerate) appear.
For implementation:
Only the diamond mesh and the measures of elements are relevant !
The matrix of the DDFV method is assembled “per diamond”.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Centers of diamonds, elements and DDFV volumes
D
Simplest choice of diamond centers xD : the diagonals’ intersection
M
M∗
(Main) example of diamond, primal and dual meshes:
xD taken at diagonals’ intersection.
NB: another natural choice: xD the barycenter of vertices of D.
(easy to calculate the measures of elements from given vertices of D).
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Centers of diamonds, elements and DDFV volumes
Primal mesh-oriented DDFV
The 2D DDFV scheme with xD at diagonals’ intersection
is the original one: Hermeline ’98,’00 , Domelevo, Omnès ’05 .
Construction from a given primal mesh M:
natural if a mesh is given (⇒ DDFV = cell+vertex - centered scheme)
xK ∗
boundary dual
volume K ∗
primal
volume
xK
K
boundary primal
“volume” K
xK
xK ∗
dual
volume
K∗
diamond
D
=union of 4 elements
Construction from diamonds:
most natural from the GS viewpoint Droniou, Eymard, Herbin ’15
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Centers of diamonds, elements and DDFV volumes
“Almost arbitrary” primal meshes allowed
Example of a primal mesh allowing for the 2D DDFV construction:
D
M
Some conditions on families Th of primal meshes are needed
such as uniform lower bound on angles (αD )D∈Dh
(condition satisfied for the meshes like those pictured above).
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
2D DDFV SCHEME
Applications of DDFV schemes
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Scalar products and Discrete Duality
Discrete functions and fields. Discrete gradient and divergence
operators
2D DDFV triple T = diamond mesh D + volumes meshes M, M∗
Consider two kinds of objects:
∗
~ T ∈ R# D 2 :
discrete functions u T ∈ R#M+#M , discrete fields F
~T = F
~D
u T = uK K ∈M , uK ∗ K ∗ ∈M∗ , F
.
D ∈D
Discrete gradient operator:
the DDFV per diamond reconstruction from opposite vertices
~ T : R#M+#M∗ −→ R#D d ,
∇
~ D u T :=
∇
1
sin αD
uL −uK ~
∗ ∗
dK ,L nK ,L
+
uL∗ −uK ∗
~
dK ∗ ,L∗ nK ,L
Discrete divergence operator: standard FV discretization
per primal/dual volume integration + Green-Gauss
d
∗
divT : R#D −→ R#M+#M ,
~ T := 1 P |∂ K ∩ D|F
~D · ~nK divK ∗ F
~ T :=
divK F
D
|K |
1
|K ∗ |
P
D
~D · ~nK ∗
|∂ K ∗ ∩ D|F
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Scalar products and Discrete Duality
Scalar products and Discrete Duality
Inner product on the space of discrete functions:
hh
ii
X
1 X
|K |uK vK +
|
K ∗ |uK ∗ vK ∗
u T , v T :=
K ∈M
K ∗ ∈M∗
2
NB: The GS viewpoint will be:
hh
ii
X
uT, v T
:=
GS
K ∈M, K ∈M∗
|K ∩ K ∗ |
uK + uK ∗ vK + vK ∗
2
2
inner product on the space of discrete fields:
o
o
n
n
X
~D · G~D
~ T , G~ T :=
|D |F
F
D ∈D
Discrete Duality (DD) property :
∗
~ T ∈ R#D
∀ u T ∈ R#M+#M ∀ F
hh
ii n
n
o
o
~ T + ∇u
~ T, F
~T = 0
u T , divT F
or =
DD
EE
..., ...
∂Ω
.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Scalar products and Discrete Duality
Calculation of discrete gradient and proof of the Discrete Duality.
~ D uT:
It is useful (both for proofs and for implementation) to rewrite ∇
~ K L + (uL∗ − uK ∗ )N
~ K ∗ L∗ ,
~ D u T = 1 (uL − uK )N
∇
2|D|
Z
Z
~ K L :=
~ K ∗ L∗ :=
~nK , N
~nK
N
K|L∩D
K|L∩D
being the quantities that also appear in discrete divergence: e.g.,
X
~ KL
~ T := 1
~D · N
F
divK F
D : D ∩∂ K 6=∅
|K |
Lemma (2D DDFV has Discrete Duality property, Dirichlet BC)
Let u T be a discrete function with zero entries in boundary volumes,
~ T de a discrete field. Then
and F
hh
ii n
n
o
o
~ T + ∇u
~ T, F
~ T = 0.
u T , divT F
hh ii
~ K L writing, gather terms with F
~D per
Proof: Write ·, · using the N
diamond, use conservativity of fluxes (summation-by-parts idea).
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Scalar products and Discrete Duality
Lift of the discrete solution to Ω. Relation to gradient schemes.
~F
~ T )(x) := P F
~D 11D (x)
The discrete fields are lifted to Ω by (Π
D
For the reasons that will become clear while studying asymptotic
compactness, we have to lift u T on Ω by
P
P
P
u +u ∗
(Πu T )(x) := 12
uK 11K (x)+ K ∗ uK ∗ 11K ∗ (x) ≡ K ,K ∗ K 2 K 11K ∩K ∗ (x).
K
Relation to gradient schemes: in the GS setting, one uses
Z
hh
ii
uT, v T
:= (Πu T )(x)(Πu T )(x) dx 6= the DDFV scalar product.
GS
Yet if v
T
Ω
is the discretization of v ∈ C 1 (Ω),
∀x ∈ K ∩ K ∗ |vK −
vK+vK ∗
2
|, |vK ∗ −
vK+vK ∗
2
| ≤ diam(K ) ⇒
hh
ii
P
u +u ∗ v +v ∗
uT, v T
= K ∈M, K ∈M∗ |K ∩ K ∗ | K 2 K K 2 K
GS
P
P
v +v ∗
v +v ∗ = 12
|K ∩ K ∗ |uK K 2 K + K ∈M, K ∈M∗ |K ∩ K ∗ |uK ∗ K 2 K
K ∈M, K ∈M∗
hh T T ii
P
P
∗
∗ vK ∗
≈ 12
|
K
|u
v
+
|
K
|u
= u ,v
∗
∗
K
K
K
K ∈M
K ∈M
up to a term of order size(T)ku T kL1 .
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Scalar products and Discrete Duality
Lift of the discrete solution to Ω. Relation to gradient schemes.
~F
~ T )(x) := P F
~D 11D (x)
The discrete fields are lifted to Ω by (Π
D
For the reasons that will become clear while studying asymptotic
compactness, we have to lift u T on Ω by
P
P
P
u +u ∗
(Πu T )(x) := 12
uK 11K (x)+ K ∗ uK ∗ 11K ∗ (x) ≡ K ,K ∗ K 2 K 11K ∩K ∗ (x).
K
Relation to gradient schemes: in the GS setting, one uses
Z
hh
ii
uT, v T
:= (Πu T )(x)(Πu T )(x) dx 6= the DDFV scalar product.
GS
Yet if v
T
Ω
is the discretization of v ∈ C 1 (Ω),
∀x ∈ K ∩ K ∗ |vK −
vK+vK ∗
2
|, |vK ∗ −
vK+vK ∗
2
| ≤ diam(K ) ⇒
hh
ii
P
u +u ∗ v +v ∗
uT, v T
= K ∈M, K ∈M∗ |K ∩ K ∗ | K 2 K K 2 K
GS
P
P
v +v ∗
v +v ∗ = 12
|K ∩ K ∗ |uK K 2 K + K ∈M, K ∈M∗ |K ∩ K ∗ |uK ∗ K 2 K
K ∈M, K ∈M∗
hh T T ii
P
P
∗
∗ vK ∗
≈ 12
|
K
|u
v
+
|
K
|u
= u ,v
∗
∗
K
K
K
K ∈M
K ∈M
up to a term of order size(T)ku T kL1 .
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Discrete compactness and lift of discrete functions.
Introduce primal and dual lift of discrete functions:
P
P
Πo u T := K uK 11K (x), Π∗ u T := K ∗ uK ∗ 11K ∗ (x)
⇒
Πu T =
Πo u T +Π∗ u T
2
Using the DD property + ad hoc consistency properties, one proves
Proposition (Discrete asymptotic compactness)
Consider a sequence of meshes Th with h = size(Th ) → 0. Assume
~∇
~ Th u Th )h are bounded in L1+sthg (Ω).
(Πo u Th )h , (Π∗ u Th )h and (Π
Then there exist u o , u ∗ such that
Πo u Th → u o , Π∗ u Th → u ∗ in L1 (Ω)
~∇
~ Th u Th *∇
~ u o +u ∗ weakly in L1 (Ω).
moreover, Π
2
⇒ this is the reason to fix reconstruction Π !
(cf. also GS analysis of Droniou, Eymard, Herbin ’15 )
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Handling of nonlinearities (reaction terms,...). Penalization.
Nonlinearities I: for b(u T ), use the GS reconstruction “per K ∩ K ∗ ”.
I.e., do not use the lumped approximations “b(u T ) ∼ b(uK ) on K ”
u +u ∗
but use “b(u T ) ∼ b( K 2 K ) on K ∩ K ∗ ” (natural for GS ideology).
Nonlinearities II: alternatively, add penalization
and keep the natural DDFV (lumped) approximations for b(u T ).
~ T u T the extra diffusion
Penalization operator: add to −divK K∇
P
uK −uK ∗
(P T u T )K := |K1 | K ∗ |K ∩ K ∗ | size(T)
.
(cf. S. Krell ’s stabilization in DDFV for Stokes pb.)
Lemma (contributions of the penalization operator)
(i) Assume v T has zero DOF at boundary nodes. Then
hh
ii P
(u −uK ∗ )(vK −vK ∗ )
P T u T , v T = K ,K ∗ K size(T)
(ii) If P T u T , u T ≤ C uniformly w.r.t h = size(Th ) → 0,
then u o = limh Πo u Th and u ∗ = limh Π∗ u Th coincide.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Handling of nonlinearities (reaction terms,...). Penalization.
Nonlinearities I: for b(u T ), use the GS reconstruction “per K ∩ K ∗ ”.
I.e., do not use the lumped approximations “b(u T ) ∼ b(uK ) on K ”
u +u ∗
but use “b(u T ) ∼ b( K 2 K ) on K ∩ K ∗ ” (natural for GS ideology).
Nonlinearities II: alternatively, add penalization
and keep the natural DDFV (lumped) approximations for b(u T ).
~ T u T the extra diffusion
Penalization operator: add to −divK K∇
P
uK −uK ∗
(P T u T )K := |K1 | K ∗ |K ∩ K ∗ | size(T)
.
(cf. S. Krell ’s stabilization in DDFV for Stokes pb.)
Lemma (contributions of the penalization operator)
(i) Assume v T has zero DOF at boundary nodes. Then
hh
ii P
(u −uK ∗ )(vK −vK ∗ )
P T u T , v T = K ,K ∗ K size(T)
(ii) If P T u T , u T ≤ C uniformly w.r.t h = size(Th ) → 0,
then u o = limh Πo u Th and u ∗ = limh Π∗ u Th coincide.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Discrete Poincare, Sobolev inequalities. Discrete Korn inequality, etc.
Many classical functional inequalities, etc.
are transposed to the discrete DDFV setting.
Poincaré-Friedrichs:
A., Boyer, Hubert ’07 (weakest assumptions on T)
Poincaré-Wirtinger, Sobolev:
A., Bendahmane, Karlsen, Pierre ’11 , Omnès, Le ’14 ,
Bessemoulin-Chatard, Chainais, Filbet ’14
NB: The two meshes are treated separately (like TPFA)
Korn inequality:
Delcourte, Omnès – Krell , strongly uses DD property
tools for Stokes and elasticity:
~ ~u for vector-valued ~u
duality formulas involving ∇
Delcourte, Domelevo, Omnès – Krell
inf − sup stability: Boyer, Krell, Nabet ’15
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Discrete Poincare, Sobolev inequalities. Discrete Korn inequality, etc.
Many classical functional inequalities, etc.
are transposed to the discrete DDFV setting.
Poincaré-Friedrichs:
A., Boyer, Hubert ’07 (weakest assumptions on T)
Poincaré-Wirtinger, Sobolev:
A., Bendahmane, Karlsen, Pierre ’11 , Omnès, Le ’14 ,
Bessemoulin-Chatard, Chainais, Filbet ’14
NB: The two meshes are treated separately (like TPFA)
Korn inequality:
Delcourte, Omnès – Krell , strongly uses DD property
tools for Stokes and elasticity:
~ ~u for vector-valued ~u
duality formulas involving ∇
Delcourte, Domelevo, Omnès – Krell
inf − sup stability: Boyer, Krell, Nabet ’15
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Maximum principle and entropy compatibility. 2D m-DDFV
Maximum principle: true on orthogonal meshes.
E.g., primal Delaunay triangulation and dual Voronoï mesh;
cartesian meshes (relevant for image processing)
(hyperbolic problems) compatibility with entropy inequalities: true
on orthogonal meshes A., Bendahmane, Karlsen ’11 .
Theoretical convergence orders:
order h, for linear problems with smooth K Domelevo, Omnès ’05
~ and piecewise
possibility of keeping order h for −div K∇u
constant K: m-DDFV extension of DDFV Boyer, Hubert ’08
handling domain decomposition Boyer, Hubert, Krell, Gander
1
order hmin{p−1, p−1 } for p-laplacian A., Boyer, Hubert ’07
h2 on uniform cartesian meshes ? Cf. A., Boyer, Hubert ’06
Numerical convergence:
in practice, particularly good approximation of gradients
orders between h and h2 for the solutions, depending on meshes
Details: Herbin, Hubert FVCA5 benchmark in 2D .
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Some properties of DDFV operators and schemes
Maximum principle and entropy compatibility. 2D m-DDFV
Maximum principle: true on orthogonal meshes.
E.g., primal Delaunay triangulation and dual Voronoï mesh;
cartesian meshes (relevant for image processing)
(hyperbolic problems) compatibility with entropy inequalities: true
on orthogonal meshes A., Bendahmane, Karlsen ’11 .
Theoretical convergence orders:
order h, for linear problems with smooth K Domelevo, Omnès ’05
~ and piecewise
possibility of keeping order h for −div K∇u
constant K: m-DDFV extension of DDFV Boyer, Hubert ’08
handling domain decomposition Boyer, Hubert, Krell, Gander
1
order hmin{p−1, p−1 } for p-laplacian A., Boyer, Hubert ’07
h2 on uniform cartesian meshes ? Cf. A., Boyer, Hubert ’06
Numerical convergence:
in practice, particularly good approximation of gradients
orders between h and h2 for the solutions, depending on meshes
Details: Herbin, Hubert FVCA5 benchmark in 2D .
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
3D C E V E -DDFV SCHEME
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
Construction starting from a primal mesh
A first 3D construction (Pierre – Hermeline — A. et al. )
uses primal+dual mesh
primal
volume
xK
K
xK
element
xK ∗
xK ∗
TKK∗,L∗|K ∗
xM ∗
xM∗
xL∗|K ∗
xK|L
xL∗
xL
xL
xL∗
interface
primal
volume
K|L
L
3D CeVe-DDFV gradient:
Diamond
D K,L
xK
xK ∗
xM∗
xL
xL∗
Gradient reconstruction: one direction from xK , xL
and two directions from the vertices of K|L.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
Gradient approximation on CeVe diamonds
xK
If the face K|L is a triangle D = DKK,∗L,L∗,M ∗ ,
the reconstruction of all the components
of the gradient is obvious:
∇T u T |D = the vector of R3 with the projections
xK ∗
xM∗
xL











And what if it is a general polygon ?
(e.g., a quadrilateral, as for cartesian primal mesh)?
uL −uK
dKL
xL∗
,
uL∗ −uK ∗
dK ∗L∗ ,
uK ∗ −uM ∗
dM ∗K ∗ ,
−−→
on xK xL
−−−→
on xK ∗ xL∗
−−−−→
on xM ∗ xK ∗
etc.
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
Back to the 2D co-volume reconstruction property
Polygon D⊂Θ, oriented by ~n⊥Θ
x4∗
signed
area |D 4,5 |
area
∗
x3
xD∗
~n
−−
→
xD∗ x1∗,2
x5∗ ≡ x1∗
|D2,3 |
→
→
− −−
n ×xD∗ x1∗,2
x1,∗2
x2∗
Let Θ be a plane in R3 with a unit normal
vector ~n, and D ⊂ Θ be a polygon.
Introduce the vertices xi∗ , i = 1, . . . , `
(numbered counter-clockwise w.r.t. the
orientation of Θ induced by ~n).
Denote the Parea of σ by |D|, we
`
have |D| =
i=1 |D i,i+1 | (sub-areas are
signed).
Let xD∗ ∈ Θ be a distinguished point .
Take xi,i∗+1 the midpoints of the edges .
Lemma
`
`
−−−→
−−−→
2 X
1 X
∗
0
(~r · xi∗ xi+
n × xσ∗ xi,i∗+1 ≡
|Di,i+1 |(~r · ~ei,i+1 )~ei,i+
1) ~
1,
|D |
|D |
i=1
i=1
−−−→
−−
→
−−−→ −−−→
∗
∗−
∗
0
:= xi∗ xi+
ei,i+
n × xσ∗ xi,i∗+1 /k ~n × xσ∗ xi,i∗+1 k.
1 /kxi xi+1 k and ~
1 := ~
For all ~r k Θ, ~r =
where
~ei,i+1
The formula can be derived from the “magical formula” of Droniou, Eymard .
NB: if ` > 3, this is one of infinitely many affine reconstruction formulas !
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
The formula (contd )
Corollary (Consistency of the gradient reconstruction)
∗
:= w1∗ . If wi∗ are the values of an affine
Take (wi∗ )`i=1 ⊂ R, w`+1
function w at the vertices xi∗ of the polygon σ, then
X`
−−−→
w ∗ − wi∗ 0
2 X̀
∗
~ = 1
~ei,i+1 ,
∇w
(wi+1
−wi∗ ) ~n × xD∗ xi,i∗+1 ≡
|Di,i+1 | i+1
i=1
|D |
|D|
di,i+1
i=1
where
−−−→
∗
0
di,i+1 := kxi∗ xi+1
k and ~ei,i+1
signed
area |D 4,5 |
x4∗
area
x3∗
∗
xD
|D2,3 |
→
→
− −−
n ×xD∗x1∗,2
~n
−−
→
xD∗ x1∗,2
x5∗ ≡ x1∗
−−−→ −−−→
:= ~n × xσ∗ xi,i∗+1 /k ~n × xσ∗ xi,i∗+1 k
x1,∗2
x2∗
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
Coercivity or lack of coercivity of CeVe-DDFV
There are infinitely many possibilities of reconstructing tangential
~ D . One leads to Discrete Duality property.
components of ∇
The 3D CeVe-DDFV gradient reconstruction:
use the 2D co-volume reconstruction (cf. Part I of these lectures)
in the polygon K|L which vertices are dual centers xK ∗ , xL∗ , xM ∗ , ...
Coercivity concern:
~ T reduced to constants, i.e.,
is the kernel of ∇
~ T ≡ ~0
∇
⇒ Πo u T ≡ const, Π∗ u T ≡ const ?
Yes, if ` = 3 , e.g. for tetrahedral primal meshes.
Yes, on cartesian meshes (` = 4) or topologically equivalent to them.
No coercivity, in general .
NB: 3D CeVeFE-DDFV scheme fixes the coercivity issue
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A primal mesh-oriented construction
Coercivity or lack of coercivity of CeVe-DDFV
There are infinitely many possibilities of reconstructing tangential
~ D . One leads to Discrete Duality property.
components of ∇
The 3D CeVe-DDFV gradient reconstruction:
use the 2D co-volume reconstruction (cf. Part I of these lectures)
in the polygon K|L which vertices are dual centers xK ∗ , xL∗ , xM ∗ , ...
Coercivity concern:
~ T reduced to constants, i.e.,
is the kernel of ∇
~ T ≡ ~0
∇
⇒ Πo u T ≡ const, Π∗ u T ≡ const ?
Yes, if ` = 3 , e.g. for tetrahedral primal meshes.
Yes, on cartesian meshes (` = 4) or topologically equivalent to them.
No coercivity, in general .
NB: 3D CeVeFE-DDFV scheme fixes the coercivity issue
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
3D CeVe-DDFV: scalar products, discrete duality and reconstruction.
Scalar products, discrete duality, reconstruction.
~ T , G~ T ∈ (R#D )3 ,
For 3D discrete fields F
o
o
n
n
~D · G~D
~ T , G~ T = P
mD F
F
D ∈D
∗
For 3D discrete functions u T , v T ∈ R#M+#M ,
hh
ii
P
P
u T , v T = 31 K ∈M mK uK vK + 32 K ∗ ∈M∗ mK ∗ uK ∗ vK ∗ ;
Lift of discrete functions in 3D CeVE-DDFV:
Πu T := 13 Πo u T + 32 Π∗ u T ,
P
P
Πo := u T (x) K ∈M uK 11K (x), Π∗ u T (x) := K ∗ ∈M∗ uK ∗ 11K ∗ (x)
Proposition (3D CeVe Discrete Duality property, Dirichlet BC)
hh
ii n
n
o
o
~ T] , uT + F
~ T , ∇T u T = 0
divT [F
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
3D C E V E FE-DDFV SCHEME
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
An alternative: 3D CeVeFE-DDFV aka “new DDFV” scheme
There is an alternative to 3D CeVe-DDFV which is always coercive.
Moreover, the construction can naturally be started from an arbitrary
octahedral diamond mesh.
3D octahedron has 3 pairs of “opposite” vertices per diamond .
The vertices are sorted pairwise into three classes and
each pair of vertices is used to reconstruct a direction of the gradient .
Each of the three types of the vertices determines a partition of Ω
⇒ CeVeFE-DDFV involves integration on three meshes
Differences w.r.t. CeVe-DDFV:
meshes weights 13 (primal), 32 (dual) are replaced by
CeVeFE scalar product:
1 1 1
3,3,3.
hh
ii
1 X
1
1 X
uT, v T =
mK uK vK +
mK ∗ uK ∗ vK ∗ + M# mesh term
3 K ∈M
3 ∗ ∗
3
K
∈M
CeVeFE reconstruction of solution:
Πu T := 31 Πo u T + 13 Π∗ u T + 13 Π# u T
Details: the “Part-III.pdf” (by the courtesy of F. Hubert)
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
An alternative: 3D CeVeFE-DDFV aka “new DDFV” scheme
There is an alternative to 3D CeVe-DDFV which is always coercive.
Moreover, the construction can naturally be started from an arbitrary
octahedral diamond mesh.
3D octahedron has 3 pairs of “opposite” vertices per diamond .
The vertices are sorted pairwise into three classes and
each pair of vertices is used to reconstruct a direction of the gradient .
Each of the three types of the vertices determines a partition of Ω
⇒ CeVeFE-DDFV involves integration on three meshes
Differences w.r.t. CeVe-DDFV:
meshes weights 13 (primal), 32 (dual) are replaced by
CeVeFE scalar product:
1 1 1
3,3,3.
hh
ii
1 X
1
1 X
uT, v T =
mK uK vK +
mK ∗ uK ∗ vK ∗ + M# mesh term
3 K ∈M
3 ∗ ∗
3
K
∈M
CeVeFE reconstruction of solution:
Πu T := 31 Πo u T + 13 Π∗ u T + 13 Π# u T
Details: the “Part-III.pdf” (by the courtesy of F. Hubert)
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
An alternative: 3D CeVeFE-DDFV aka “new DDFV” scheme
There is an alternative to 3D CeVe-DDFV which is always coercive.
Moreover, the construction can naturally be started from an arbitrary
octahedral diamond mesh.
3D octahedron has 3 pairs of “opposite” vertices per diamond .
The vertices are sorted pairwise into three classes and
each pair of vertices is used to reconstruct a direction of the gradient .
Each of the three types of the vertices determines a partition of Ω
⇒ CeVeFE-DDFV involves integration on three meshes
Differences w.r.t. CeVe-DDFV:
meshes weights 13 (primal), 32 (dual) are replaced by
CeVeFE scalar product:
1 1 1
3,3,3.
hh
ii
1 X
1
1 X
uT, v T =
mK uK vK +
mK ∗ uK ∗ vK ∗ + M# mesh term
3 K ∈M
3 ∗ ∗
3
K
∈M
CeVeFE reconstruction of solution:
Πu T := 31 Πo u T + 13 Π∗ u T + 13 Π# u T
Details: the “Part-III.pdf” (by the courtesy of F. Hubert)
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
A PPLICATIONS OF DDFV SCHEMES
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Problems discretized with DDFV
Linear anisotropic and heterogeneous elliptic equations:
Hermeline – Domelevo, Omnès – ...
Flows in porous media:
Boyer, Hubert – Chainais, Krell, Mouton – ...
Nonlinear elliptic and elliptic-parabolic equations:
A., Boyer, Hubert – Coudière, Hubert – A., Bendahmane, Hubert
Nonlinear convection-diffusion equations:
Coudière, Manzini – A., Bendahmane, Karlsen
Electrocardiology (elliptic-parabolic reaction-diffusion system):
Pierre – Coudière, Pierre, Turpault – A., Bendahmane, Karlsen, Pierre
Stokes and Navier-Stokes problems:
Delcourte, Domelevo, Omnès – Krell – Krell, Manzini – Goudon, Krell –
Le, Omnès – Delcourte, Omnès – Boyer, Krell, Nabet –...
Linear elasticity:
F. Pascal, B. Martin
Image restoration, level-set curvature driven eqn.:
Handlovičová, Kotorová – Handlovičová, Frolkovič – Hartung, Hubert
2D diamond partition and gradient
2D DDFV scheme
The 3D CeVe-DDFV scheme
The 3D CeVeFE-DDFV scheme
Applications of DDFV schemes
Thank you !
That’s all... thank you — merci !