Mimetic Finite Difference methodsAn introduction
Andrea Cangiani
Universita di Roma La Sapienza
Seminario di Modellistica Differenziale Numerica – 2 dicembre 2008
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 1 / 30
Some cronology...
RECENTLY (1995–2001)
Shashkov-Steinberg, JCP 1995 (support-operator methods)
(Hyman)-Shashkov-Steinberg, JCP 1996-(7) (rough diffusion)
Hyman-Shashkov, SINUM 1999 (Maxwell)
Campbell-Shashkov, JCP 2001 (gas dynamics)
MORE RECENTLY (2004–2008)
Kuznetsov-Lipnikov-Shashkov, Comp. Geos. 2004 (polygons)
Brezzi-Lipnikov-Shashkov, SINUM 2005 (error analysis)
Brezzi-Lipnikov-Simoncini, M3AS 2005 (a new family of MFD)
Brezzi-Lipnikov-Shashkov, M3AS 2006 (curved faces)
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 2 / 30
Some cronology (continued)
MORE RECENTLY (2004–2008) – continued
Beirao Da Veiga, NM 2007 (residual error estimator)
C.-Manzini, CMAME 2008 (post-processing)
C.-Manzini-Russo, SINUM accepted (convection-reaction-diffusion)
IN PROGRESS
Convection-dominated diffusion
Higher-order MFD
Nodal MFD
Mimetic curl operator
Mimetic discretization of Stokes
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 3 / 30
Features
RECENTLY
Generalisation of finite differences to hexahedral meshes.
Discrete differential operators defined so as to mimic the properties of theunderlying continuum operators (e.g. vector calculus identities, conservationlaws, solution symmetries)
Applied to wide range of problems.
MORE RECENTLY
Generalisation of (low order) Mixed Finite Elements/finite volumes togeneral polyhedral meshes. Constructs a family of methods.
Discrete differential operators defined so as to mimic the properties of theunderlying continuum operators. May gain from extra freedom given bymethod construction
Currently limited to linear diffusion problems
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 4 / 30
Compatible Spatial Discretizations
We define compatible spatial discretizations as those that inherit or mimicfundamental properties of the PDE such as topology, conservation,symmetries, and positivity structures and maximum principles.(IMA ”Hot Topics” Workshop Compatible Spatial Discretizations forPartial Differential Equations, May 11-15, 2004)
Advantages:
Conserve crucial features of physical, geometrical, and mathematicalmodel:
I Conservation lawsI SymmetryI Positivity and monotonicityI Duality properties of differential operators
Provide reliability and accuracy
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 5 / 30
Compatible Spatial Discretizations
The following can be cast as compatible spatial discretizations:
mixed finite element methods, mimetic finite differences, support operatormethods, control volume methods, discrete differential forms, Whitneyforms, conservative differencing, discrete Hodge operators, discreteHelmholtz decomposition, finite integration techniques, staggered grid anddual grid methods, etc.
Support operator method:
Most PDEs are written in terms of invariant differential operators(div, grad, and curl)
Define discrete analogues of these invariant operators that satisfyexacly the discrete analogs of the identities satisfied by the continuumoperators (e.g. Gauss’, Stokes’, Hodge orthogonal decomposition).
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 6 / 30
Linear diffusion in mixed form
Consider the linear diffusion equation bvp
−div(Kgrad p) = b in Ω ⊂ IR2,3
p = 0 on ∂Ω
with K strongly elliptic full symmetric tensor.
Mixed formulation:
F = −K grad p (Constitutive Equation)
div F = b (Conservation Equation)
We also introduce the flux operator
Gp = −Kgrad p.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 7 / 30
Linear diffusion in mixed form
The operators div and grad satisfy the integral identity∫Ω
p divF dV +
∫Ω
F · gradp dV = 0
which we rewrite: ∫Ω
p divF dV −∫
ΩK−1F · Gp dV = 0 (1)
Introducing the inner-products:
(p, q)W =∫
Ω p q dV
(F ,G )V =∫
Ω K−1F · G dV
the identity (1) becomes:
(divF , p)W − (F ,Gp)V = 0
expressing the fact that divergence and flux are adjoint: G = grad∗Andrea Cangiani (IAC–CNR) mimetic finite difference methods 8 / 30
Linear diffusion in mixed form
Consider the linear diffusion equation bvp
−div(Kgrad p) = b in Ω ⊂ IR2,3
p = 0 on ∂Ω
with K strongly elliptic full symmetric tensor.
Mixed formulation:
F = −K grad p (Constitutive Equation)
div F = b (Conservation Equation)
Mixed variational formulation: find (p,F ) ∈W × V s.t.
(K−1F ,G )− (p,divG ) = 0 ∀G ∈ V
(divF , q) = (b, q) ∀q ∈W
W = L2(Ω), V = H(div; Ω)Andrea Cangiani (IAC–CNR) mimetic finite difference methods 9 / 30
Mixed Finite Element RT0-P0 Discretisation
Ω
E
E
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 10 / 30
Mixed Finite Element RT0-P0 Discretisation
Let Ωh be a triangularization of Ω ⊂ IR2. For E ∈ Ωh, let
P0(E ) = Polynomials of degree zero on E
RT 0(E ) = a + bx : a ∈ (P0(E ))2, b ∈ P0(E )
from which we define the conforming FE spaces:
Wh = qh : qh|E ∈ P0(E ), ∀E ∈ Ωh
Vh = Gh : Gh|E ∈ RT 0(E ) and Gh · n cont. on ∂Ωh
MFE RT0-P0 discretization: find (ph,F h) ∈Wh × Vh s.t.
(K−1F h,Gh)− (ph, divGh) = 0 ∀Gh ∈ Vh
(divF h, qh) = (b, qh) ∀qh ∈Wh
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 11 / 30
RT0-P0 nodal variables
Finite Element Nodal variables Lifting(E ,P0,N0) N0(qh) = qE LE
0 qE ∈ P0
qE := 1|E |∫E
q LE0 qE ≡ qE
(E ,RT 0,NRT ) NRT (G h) = (G eE )e = GE LE
RT GE ∈ RT 0
G eE := 1
|e|∫e
G · neE LE
RT GE |e · neE = G e
E
Global FE space Global nodal spaces Global lifting
Wh Qh := q = (qE )E∈Ωh L0 : Qh →Wh
Vh Xh :=
G = (GE )E∈Ωh
:
G eE−
+ G eE+
= 0 ∀e ∈ ∂Ωh
LRT : Xh → Vh.
Where E− and E+ are the two elements sharing the edge.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 12 / 30
Discrete divergence
Let LRT G = Gh. Then,
|E | div Gh|E =
∫E
divGhGauss
=
∫∂E
G · nE =∑e∈∂E
|e|G eE
⇒ divLERT G = divGh|E =
1
|E |∑e∈∂E
|e|G eE =: divhGE
We may now define the discrete divergence operator
divh : Xh → Qh
G→ divhG = (divhGE )E
satisfying divLRT G = divhG.
⇒ (ph,div Gh) =∑E
|E |pEdivhGE = [p,divhG]Qh
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 13 / 30
Scalar product in Xh
(K−1F h,Gh) =
∫Ω
K−1LRT F · LRT G
=∑E
∫E
K−1LERT FE · LE
RT GE
=∑E
[FE ,GE ]E =: [F,G]Xh
MFE in terms of the spaces Qh and Xh: find (p,F) ∈ Qh × Xh:
[F,G]Xh− [p, divhG]Qh
= 0 ∀G ∈ Xh
[divhF,q]Qh= [b,q]Qh
∀q ∈ Qh.
with b = bI, I =interpolation operator defined for q ∈ V by
(qI)E :=1
|E |
∫E
q ∀E ∈ Ωh
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 14 / 30
Mimetic finite difference discretization
Ωh partition of Ω into polygonal (polyhedral) elements.
Ω
e
E
E
qE
GE
Why polyhedral meshes?
They naturally arise in the treatment of complex solution domainsand heterogeneous materials (e.g. reservoir models)
They facilitate adaptive mesh refinement/de-refinement
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 15 / 30
Discrete MFD pressure and flux spaces
Define the discrete pressure space Qh and flux space Xh.
q ∈ Qh → q = (qE )E∈Ωh(⇒ piecewise constant)
G ∈ Xh → G =(G e
E
)e∈∂E
E∈Ωh(⇒ constant normal component)
e ∈ ∂E− ∩ ∂E+ ⇒ F eE− + F e
E+= 0
We have the interpolation operators:
For q ∈W , (qI)E = 1|E |∫E q ∀E ∈ Ωh,
For G ∈ V ,(G
I)eE
= 1|e|∫e G · ne
E ∀E ∈ Ωh ∀e ∈ ∂E
And the discrete divergence operator:
divh : Xh → Qh
G→ divhG = (divhGE )E divhGE =1
|E |∑e∈∂E
|e|G eE
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 16 / 30
Discretisation of the conservation equation
Let b = bI. The mimetic discrete conservation equation reads:
divhF = b
i.e.1
|E |∑e∈∂E
|e|FeE = (b)E ∀E (2)
Equivalently,[divhF,q]Qh
= [b,q]Qh∀q ∈ Qh
where the scalar product for the pressure space Qh is given by
[p,q]Qh=∑
E∈Ωh
|E |pE qE
This MFD discrete conservation equation (2) coincides with the MFERT0-P0 and also with many finite volume discretisation of theconservation equation.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 17 / 30
Finite volume discretisation of the constitutive equation
Directly discretize the constitutive equation defining the flux:
F = −K grad p (with K = k I)
LnAB
B
A
K
L
B
A
K
−nAB · F ≈ kpK − pL
|K− L|− F ≈ k
(pK − pL
|K− L|nAB
γAB+
pB − pA
|B− A|nKL
γKL
)
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 18 / 30
Mimetic discretisation of the constitutive equation
Introduce a scalar product for the flux space Xh
[F,G]Xh=∑
E∈Ωh
[F,G]E . ([·, ·]E to be defined!).
Moreover, define a discrete flux operator Gh : Qh → Xh as the adjoint ofdivh:
[G,Ghq]Xh= [q, divhG]Qh
, ∀q ∈ Qh ∀G ∈ Xh.
This definition naturally establishes a discrete Green formula with respectto the discrete scalar products. We say that the discrete operators mimicthe continuous ones.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 19 / 30
Discretisation of the constitutive equation (cont.)
The mimetic discretisation of the constitutive equation reads:
F = Ghp
or, equivalently,
[F,G]Xh= [Ghp,G]Xh
=[p,divhG]Qh∀G ∈ Xh
The (family of) Mimetic Finite Difference (MFD) schemes:
[F,G]Xh− [p,divhG]Qh
= 0 ∀G ∈ Xh
[divhF,q]Qh= [b,q]Qh
∀q ∈ Qh
or, equivalently,[A Bt
B 0
] [Fp
]=
[0b
]Andrea Cangiani (IAC–CNR) mimetic finite difference methods 20 / 30
MFE and MFD
As we already saw, if Ωh is made of triangles (tetrahedrons),
The MFD conservation equation discretisation is equivalent toRT0-P0
If we define the Xh-scalar product to be
[F,G]Xh=
∫Ω
K−1LRT F · LRT G
the constitutive equation disretisation is equivalent.
...and the two methods coincide!
Which properties define LRT ? Locally, on each E ∈ Ωh,
1 LERT G|e · ne
E = G eE ∀e ∈ ∂E ,
2 divLERT G = divhG|E
3 LERT G
I= G if G is constant
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 21 / 30
The flux scalar product: P0-compatible liftings
IDEA: any reasonable lifting can be used to define a scalar productyielding a reasonable MFD formulation.
Set XE = Xh|E . We define P0-compatible lifting a linear mapLE : XE → L2(E ) such that for all G ∈ XE ,
1. LEG|e · neE = G e
E , ∀e ∈ ∂E ,
2. divLEG = divhG|E ,
and, for all constant vector C ,
3. LE CI
= C .
The following defines a MFD scalar product for the flux space:
[F,G]E =
∫Ω
K−1LEF · LEG dS .
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 22 / 30
The flux scalar product: P0-compatible liftings
Assume that K is constant on E .A crucial property. For any linear function q1 and G ∈ XE
[(K∇q1)I,G]E =
∫E
K−1LE((K∇q1)I
)· LEG dS
3=
∫E
∇q1 · LEG dS
Green= −
∫E
q1divLEG dS +
∫∂E
q1(LEG · next) dl
2,1= −
∫E
q1divhG dS +∑e∈∂E
G eE
∫e
q1 dl
(local consistency)
the scalar product is independent from the lifting if one of the factors is theinterpolant of a constant vector field.
the scalar product is exact on the interpolant of constant vector fields:
[(ei )I, (ej)
I]E = (K−1)i,j |E |
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 23 / 30
A general locally consistent scalar product
IDEA: Use local consistency to construct a scalar product without a lifting!
Let kE be the number of edges (faces) of E . Set:
[F,G]E =
kE∑s,t=1
Ms,tE F es
E G etE
Imposing local consistency and symmetry we get a set of positivesemi-definite matrices
Imposing stability: exists s∗, s∗ > 0 independent of h s.t.
s∗
kE∑i=1
(G eiE )2|E | ≤ [G,G]E ≤ s∗
kE∑i=1
(G eiE )2|E | ∀G ∈ XE .
we restrict to a set of acceptable (symmetric & pos. def.) matricesdefining a family of reasonable (e.g. exact on constants) scalarproducts.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 24 / 30
A family of consistent and stable MFD scalar products[Brezzi-Lipnikov-Simoncini, Math. Models Methods Appl. Sci. (2005)]
Imposing:
local consistency ⇒ M0E symmetric & semidef.
stability ⇒ ME = M0E +CUC t symmetric & pos. def., where
C is a given (computable) full-rank kE × (kE − d) matrix,U is any symmetric & pos. def. (kE − d)× (kE − d) matrix
Proposition: Under a reasonable assumption on the minimum eigenvalueof U, the matrix ME is induced by a lifting.
In practice (so far) we use:
U = u (|E |Trace(K−1)) IIkE−d , u ∈ IR
and the proposition requires that u is sufficiently large.
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 25 / 30
MFD error in function of u
10-3
10-2
10-1
100
10110
-3
10-2
10-1
100
101
102
pressure |||.|||-errorpressure L2-errorlinear pressure L2-errorflux |||.|||-error
Quadrilaterals 20x20BLS Example 1
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 26 / 30
Back to Mixed finite elementsC. & Manzini, CMAME, 2008
Example. If Ωh is made of triangles,
ME = M0E + uE
[|ei | |ej |
]i ,j.
with M0E symmetric and positive semidefinite.
xexB
uE =
∑kEi=1
(xei − xB
)TK−1
(xei − xB
)12|E |
⇐⇒ RT 0− P0
Example Rectangles → central finite differences
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 27 / 30
Assumptions on the partition
The number of edges per face and the number of faces per element isunif. bounded.
The length l of each edge is such that l ≤ Ch (shape regularity).
Every element is uniformly strictly starshaped
(if Ω ∈ IR3) Every face is uniformly strictly starshaped
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 28 / 30
Error estimates[Brezzi-Lipnikov-Shashkov, SIAM J. Num. Anal. (2005)]
If
Ω is a convex polyhedron (polygon) with Lipschitz continuousboundary,Ωh is a partition satisfying the previous slide assumptions,the scalar product satisfies local consistency and stability.p ∈ H2(Ω)
Then,
‖|pI − p‖|Qh+ ‖|F I − F‖|Xh
≤ C h,
⇒ ‖p − L0p‖L2 ≤ C h
where‖| · ‖|2Qh= [·, ·]Qh
and ‖| · ‖|2Xh= [·, ·]Xh
.
If, moreover,
The scalar product is induced by (some) lifting.
Then,
‖|pI − p‖|Qh≤ C h2 (pressure super convergence)
Andrea Cangiani (IAC–CNR) mimetic finite difference methods 29 / 30