Mimetic Finite Difference methods - The state of the art...Andrea Cangiani (IAC{CNR) mimetic nite di erence methods 11 / 41 Assumptions on the partition The number of edges per face

Mimetic Finite Difference methodsThe state of the art

Andrea Cangiani

Universita di Roma La Sapienza

Seminario di Modellistica Differenziale Numerica – 16 dicembre 2008

Andrea Cangiani (IAC–CNR) mimetic finite difference methods 1 / 41

Outline

1 MFD for linear diffusionMFD discretisationError estimates and post-processingMesh adaptivity

2 Convection-reaction-diffusionMFD discretisationError estimates

3 Convection-dominated diffusionMFD discretisationStabilisationNumerical examples

4 Higher order and nodal MFDNodal MFD


Linear diffusion in mixed form

Mixed formulation:

F = −K grad p (Constitutive Equation)

div F = b (Conservation Equation)

Mixed variational formulation: find (p,F ) ∈W × V such that

(K−1F ,G )− (p,divG ) = 0 ∀G ∈ V

(divF , q) = (b, q) ∀q ∈W

where W = L2(Ω), V = H(div; Ω).

Mimetic Finite Difference (MFD): find (p,F) ∈ Qh × Xh such that

[F,G]Xh− [p,divhG]Qh

= 0 ∀G ∈ Xh

[divhF,q]Qh= [b,q]Qh

∀q ∈ Qh

where b = bI, and for discrate spaces Qh and Xh defined below.Andrea Cangiani (IAC–CNR) mimetic finite difference methods 3 / 41

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


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


Mimetic scalar products

The scalar product for the pressure space Qh

[p,q]Qh=∑

E∈Ωh

|E |pEqE

The scalar product for the flux space Xh

[F,G]Xh=∑

E∈Ωh

[F,G]E ([·, ·]E to be defined!).

May define a discrete flux operator Gh : Qh → Xh adjoint of divh:

[G,Ghq]Xh= [q, divhG]Qh

, ∀q ∈ Qh ∀G ∈ Xh.


The flux scalar product: P0-compatible liftings

IDEA: any reasonable lifting (field reconstraction) can be used to define ascalar product yielding 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. LECI

= C .

The following defines a MFD scalar product for the flux space:

[F,G]E =

∫E

K−1LEF · LEG dS .


Mimetic scalar product: local consistency

Local consistency.Assume that K is constant on E . For every linear function q1 and G ∈ XE

we require that

[(K∇q1)I,G]E = −∫

Eq1divhG dS +

∑e∈∂E

G eE

∫eq1 dl

Let xB be the barycenter of E , and C be a constant vector field.

Given C = CI, using local consistency with q1 = K−1C · (x− xB)

[C,G]E =∑e∈∂E

G eE

∫eK−1C · (x− xB), dl

It easily follows that, the scalar product is exact on discrete fieldsassociated to constant fields. In particular, setting Gi = (ei )

I, i = 1 : d ,

[Gi ,Gj ]E = (K−1)i ,j |E | i , j = 1 : d .


Mimetic scalar product: local consistency

Change of basis for XE .Let kE be the number of edges (faces) of E , and XE = Xh|E .Consider a new basis EikE

i=1 for XE such that

Ei := Gi i = 1 : d ,

and for all i = 1 : d

[Gi , Ej ] =∑e∈∂E

Eej

∫eK−1ei · (x− xB) = 0 ∀j = d + 1 : kE .

In the new basis, the locally consistent scalar product has the form:

K−1|E | 0

0 ?


Mimetic scalar product: local consistency[Brezzi, Lipnikov, Shashkov, Simoncini, CMAME 2007]

Theorem Let S be a (kE − d)× (kE − d) s.p.d. matrix.If the smallest eigenvalue of S is above a threshold, then the scalarproduct given by the matrix

K−1|E | 0

0 S

is associated to a P0-compatible lifting.

In practice (so far) we used:

S = |E |Trace(K−1)IIkE−d


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 +CEUC t

E symmetric & pos. def., whereCE is a given (computable) full-rank kE × (kE − d) matrix,U is any symmetric & pos. def. (kE − d)× (kE − d) matrix

In practice, the matrices ME and CE are constructed translating theabove conditions in the language of linear algebra. The computational costis similar to that of finite element assembly.


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


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)


Postprocessing based on vector fields reconstruction[C. & Manzini, CMAME, 2008]

Let V be a vector field on E .

VInterpolate

−→ VI ∈ XE

Reconstruct

−→ VR ∈ (P0(E ))2

Define the (local) reconstruction operator componentwise as

(VR

)i =1

|E |

[(Kei )

I ,VI]E

(i = 1, . . . , d).

It follows that the reconstruction process

is independent of the scalar product,

is exact for constant vector fields,

gives the best constant approximation w.r.t. the norm ‖| · ‖|EMoreover, if E is a semi-regular polyhedron, then V

R= V (xBE

)


Postprocessing of the pressure field

Given the MFD solution (p,F), we

1. reconstruct a piecewise constant flux as

(FR)i =1

|E |

[(Kei )

I ,F]E

(i = 1, . . . , d), ∀E ∈ Ωh.

2. define a reconstructed piecewise linear pressure field as

pRh (x) := LE

0 pE + K−1FR · (x − xBE) ∀x ∈ E , ∀E ∈ Ωh

Theorem. Under the hypothesis yielding pressure super conv.

‖p − pRh ‖L2 ≤ C h2 (pressure L2-super convergence)


Some kind of a (dual) mesh


Pressure errorΩ = (0, 1)2, K = II, p(x , y) = x3 y 2 + x sin(2πx) sin(2πy)

Dual mesh

NE Ne hmax ‖|pI − p‖|QhRate ||p − pR

h ||L2 Rate289 1248 7.02 10−2 1.03 10−1 −− 1.20 10−1 −−

1089 4800 3.51 10−2 3.35 10−2 1.628 3.75 10−2 1.6814225 18816 1.75 10−2 9.04 10−3 1.890 1.00 10−2 1.902

16641 74496 8.77 10−3 2.32 10−3 1.959 2.57 10−3 1.96366049 296448 4.38 10−3 5.83 10−4 1.995 2.57 10−3 1.963

Structured triangular mesh

NE Ne hmax ‖|pI − p‖|QhRate ||p − pR

h ||L2 Rate200 320 7.07 10−2 3.39 10−2 −− 5.43 10−2 −−800 1240 3.53 10−2 8.66 10−3 1.970 1.38 10−2 1.976

3200 4880 1.76 10−2 2.18 10−3 1.989 3.47 10−3 1.99212800 19360 8.84 10−3 5.47 10−4 1.995 8.69 10−4 1.99751200 77120 4.42 10−3 1.37 10−4 1.998 2.17 10−4 1.999


Flux errorΩ = (0, 1)2, K = II, p(x , y) = x3 y 2 + x sin(2πx) sin(2πy)

Dual mesh

NE Ne hmax |||F I − F||| Rate289 1248 7.020 10−2 1.348 −−

1089 4800 3.510 10−2 6.631 10−1 1.0234225 18816 1.755 10−2 3.342 10−1 0.988

16641 74496 8.775 10−3 1.659 10−1 1.01066049 296448 4.387 10−3 8.323 10−2 0.994


NE Ne hmax |||F I − F||| Rate200 320 7.071 10−2 5.384 10−2 −−800 1240 3.536 10−2 1.391 10−2 1.952

3200 4880 1.768 10−2 3.513 10−3 1.98512800 19360 8.839 10−3 8.819 10−4 1.99351200 77120 4.419 10−3 2.211 10−4 1.996


Reconstructed gradient error at baricenterΩ = (0, 1)2, K = II, p(x , y) = x3 y 2 + x sin(2πx) sin(2πy)


NE Ne hmax ||(∇p)B − FB || Rate200 320 7.071 10−2 3.256 10−1 −−800 1240 3.536 10−2 1.587 10−1 1.036

3200 4880 1.768 10−2 7.881 10−2 1.00912800 19360 8.839 10−3 3.934 10−2 1.00251200 77120 4.419 10−3 1.966 10−2 1.000

Structured quadrilateral mesh

NE Ne hmax ||(∇p)B − FB || Rate400 840 6.360 10−2 1.305 10−1 −−

1600 3280 3.188 10−2 3.664 10−2 1.8326400 12960 1.595 10−2 9.508 10−3 1.946

25600 51520 7.975 10−3 2.404 10−3 1.983102400 205440 3.988 10−3 6.031 10−4 1.994


A posteriori error estimation[Beirao Da Veiga, NM, 2007 and Beirao Da Veiga & Manzini, IJNME. PDE, 2008]

The reconstructed pressure pRh is used to define an

error estimator

η2 :=∑

E∈Ωh

η2E

η2E := ‖|F + (K∇pR

h )I‖|2E +1

2

∑e∈∂E

h−1e ‖[[pR

h ]]‖20

with [[·]] is the jump across the edge.

Above and below a posteriori error bounds hold for the error

‖F I − LF‖div,h + ‖|p − pRh ‖|1,h


Mesh adaptivity

The local error indicator is used to select elements needing refinement:

1. calculate local error estimators ηE

2. mark for refinement elements such that ηE ≥ crefηmax

3. subdivide the marked elements as follows


Example of mesh adaptationΩ = (0, 1)2, K = II, u(x , y) = 16x(1− x)y(1− y)arctan(25x − 100y + 50)


A posteriori error estimator efficiency

Error and error estimator for uniformly (solid) and adaptively (dashed)refined meshes

102

103

104

105

106

100

101

102

Number of elements


A problem with highly (1/100) discontinuous K andsolution’s tangential flux at interfaces


Convection-reaction-diffusion[C. & Manzini & Russo, SINUM, subm.]

Consider the linear convection-reaction-diffusion equation

−div(Kgrad p + βp) + c p = b,

where β is a smooth velocity field, c a reaction coefficient.

Mixed formulation:F = −(K grad p + βp)

div F + c p = b.

Mixed variational formulation: find (p,F ) ∈W × V s.t.

(αF ,G )− (p,divG ) + (αβp,G ) = 0 ∀G ∈ V

(divF , q) + (cp, q) = (b, q) ∀q ∈W

where α := K−1.Andrea Cangiani (IAC–CNR) mimetic finite difference methods 25 / 41

MFD discretisation...

...of the new terms:

(αβp,G ) =∑E

∫Eαβp · G →

∑E

pE

[βI,G

]E

(c p, q) →[cIp,q

]Qh

Mimetic Finite Difference (MFD): find (p,F) ∈ Qh × Xh such that

[F,G]Xh− [p, divhG]Qh

+∑E

pE

[βI,G

]E

= 0 ∀G ∈ Xh

[divhF,q]Qh+ [cIp,q]Qh

= [b,q]Qh∀q ∈ Qh


Error estimates

Assume that the scalar product for Xh is given by a P0-compatible lifting.Then, we can re-write the MFD method in terms of L2:

(αLF,LG)− (L0p, div LG) + (αLβIL0p,LG) = 0 ∀G ∈ Xh

(div LF,L0q) + (L0(cIp),L0q) = (L0b,L0q) ∀q ∈ Qh

and analyse it as a finite-element method (a la Douglas & Roberts ’85).

Theorem. Under all assumptions seen before, and assuming h small enoughand that the scalar product of Xh is related to a P0-compatible lifting

||p − L0p||L2 + ||F − LF||H(div) ≤ C h

If, moreover, K is piecewise constant and β = LβI, then

‖|pI − p‖|Qh≤ C h2 (pressure super-convergence)


Meshes used for numerical tests

M1 M2


Pressure errorΩ = (0, 12), K = I, β = (1, 3)T , c = xy 2, and p = sin(2πx) sin(2πy) + x2 + y 2 + 1

MeshM1 MeshM2

n ||p − L0p||abs0 ||p − L0p||rel0 rate ||p − L0p||abs

0 ||p − L0p||rel0 rate

10 1.635 10−1 9.134 10−2 −− 1.449 10−1 8.095 10−2 −−20 8.290 10−2 4.630 10−2 1.007 7.098 10−2 3.964 10−2 1.029

40 4.144 10−2 2.315 10−2 1.012 3.521 10−2 1.966 10−2 1.011

80 2.083 10−2 1.164 10−2 0.995 1.756 10−2 9.810 10−3 1.003

160 1.046 10−2 5.841 10−3 0.995 8.777 10−3 4.902 10−3 1.000

320 5.241 10−3 2.927 10−3 0.996 4.388 10−3 2.451 10−3 1.000

MeshM1 MeshM2

n ‖|pI − p‖|absQh

‖|pI − p‖|relQhrate ‖|pI − p‖|abs

Qh‖|pI − p‖|relQh

rate

10 5.491 10−2 3.069 10−2 −− 3.241 10−2 1.813 10−2 −−20 1.930 10−2 1.078 10−2 1.551 9.396 10−3 5.250 10−3 1.787

40 5.026 10−3 2.807 10−3 1.964 2.439 10−3 1.362 10−3 1.946

80 1.340 10−3 7.483 10−4 1.913 6.203 10−4 3.464 10−4 1.975

160 3.409 10−4 1.904 10−4 1.975 1.561 10−4 8.720 10−5 1.990

320 8.587 10−5 4.796 10−5 1.989 3.907 10−5 2.182 10−5 1.998


Convection-dominated diffusion

Consider singularly perturbed equation

div(−εKgrad p + βp) = b,

(β smooth velocity field, ε > 0 singular perturbation parameter)

Mixed formulation:F = −εK grad p,

div (F + βp) = b.

Remark. We could also consider solving

F = −εK grad p + βp

div F = b.


MFD discretisation

Given some mapQh −→ Xh

p −→ βp =(βp

e

E

)we have the MFD method

[F,G]Xh− [p, divhG]Qh

= 0 ∀G ∈ Xh

[divhF,q]Qh+ [divhβp,q]Qh

= [b,q]Qh∀q ∈ Qh.

Or, in algebraic form, [A Bt

B C

] [Fp

]=

[0b

]After solving, we can construct a full discrete flux F = F + βp.


Stabilisation

For stability in convection-dominated regime, i.e. ε |β|,

βpe

E = (βI)eEpe− (upwind method)

Other way to stabilisation: use the average (pe− + pe

+)/2 and add thejump-penalty term ∑

e

|e|ce [[p]][[q]],

where [[p]] = pe+ − pe

−, and ce is a penalisation parameter.

A well-known fact in the Discontinuous-Galerkin community:

upwinding = jump-penalisation with ce := |β · ne |/2


Triangular mesh


Polygonal mesh


Pressure errorΩ = (0, 1)2, K = II, β = (1, 1), u(x , y) = 2 sin(x)

“1− e

1−xε

”y 2

“1− e

1−yε

”

10-3

10-2

10-1

h_max

10-3

10-2

10-1

100

Err

or

ε=1Ε0ε=1Ε−1ε=1Ε−2ε=1Ε−3ε=1Ε−4ε=1Ε−5ε=1Ε−6

Pressure |||.|||-error(Triangles)


Pressure errorΩ = (0, 1)2, K = II, β = (1, 1), u(x , y) = 2 sin(x)

“1− e

1−xε

”y 2

“1− e

1−yε

”

10-3

10-2

10-1

h_max

10-3

10-2

10-1

100

Err

or

ε=1Ε0ε=1Ε−1ε=1Ε−2ε=1Ε−3ε=1Ε−4ε=1Ε−5ε=1Ε−6

Pressure |||.|||-error(Polygonal Mesh)


New MFD methods

Variants of the basic (low order) MFD

Curved Faces Treats generalised polyhedrons. Flux represented bythree d.o.f. on curved faces[Brezzi, Lipnikov, Shaskov, M3AS ’06, + Simoncini, CMAME ’07]

Higher order MFD Linear representation of the flux.[ Gyrya, Lipnikov, JCP. in press, sa Veiga, Manzini, SISC ’08]

Other MFD methods

Nodal MFD Define MFD on −div(K grad p) = b[Brezzi, Buffa, Lipnikov, IMATI ’08]


Nodal MFD for linear diffusion[Brezzi, Buffa, Lipnikov, IMATI ’08]

Model problem Find u ∈ H10 (Ω) such that

−div(K grad u) = b in Ω.

Variational formulation: find u ∈ V = H10 (Ω) such that

(K grad u, grad q) = (b, v) ∀v ∈ V .

In order to give sense to pointwise values, define

H1(f ) = v ∈ H1(f ) such that v |∂f ∈ H1(∂f ) for each face f

H1(E ) = v ∈ H1(E ) such that v |f ∈ H1(f ) for each element E

Notice change of notations!


Nodal MFD space

Let V be the set of all vertices in Ωh. We define the space of nodalvariables Nh

u ∈ Nh → u = (uv )v∈V .

andN 0

h = u ∈ Nh such that uv = 0 ∀v ∈ ∂Ω.

We define the discrete grad operator

(GRAD u)|e = uv2 − uv1

where e is an edge of verticies v1, v2.

This maps from Nh to a space of edge variables Eh.Similarly, N 0

h is mapped into E0h .


Nodal MFD formulation

Nodal MFD formulation: find u ∈ N 0h such that

[[u, v]] := [GRAD u,GRAD v]E = (b, v)N ∀v ∈ N 0h

...once appropriate scalar mimetic products are defined!

Scalar product in Nh is (b, v)N =∑E

bE

vE∑i=1

vviωiE

where vE is the number of vertices of E , and ωiE a set of weights of a

quadrature formula exact on linears on each face.

The scalar product for the edge space Nh is defined using a localconsistency condition, similarly to classical MFD...

May define discrete DIV operator as the adjoint of the GRAD operator!


Summary

Family of MFD methods for linear diffusion and convection-diffusionI Based on definition of discrete differential operators (div-grad)

satisfying a discrete Green formula

I Very general partitions (non-convex, generalised polyhedra)

I Possible to reproduce analysis of more classical finite element methods(a priori analysis, postprocessing, aposteriori analysis, adaptivity, andeigenvalues)

OutlookI More on convection-diffusionI More on higher order MFD

I Mesh adaptivity with corsening

I MFD for: time-dependent, Maxwell (rot-grad), Stokes, ...


Documents

Mimetic Finite Difference methods - The state of the art...Andrea Cangiani (IAC{CNR) mimetic nite di erence methods 11 / 41 Assumptions on the partition The number of edges per face