Design Principles of the Mimetic Finite Difference Schemes · 2015-10-28 · Design Principles of the Mimetic Finite Di erence Schemes Konstantin Lipnikov Los Alamos National Laboratory,

Design Principles of the Mimetic Finite DifferenceSchemes

Konstantin Lipnikov

Los Alamos National Laboratory, Theoretical DivisionApplied Mathematics and Plasma Physics Group

October 2015, Georgia Tech, GA

Co-authors: L.Beirao da Veiga, F.Brezzi, V.Gyrya, G.Manzini,D.Moulton, V.Simoncini, M.Shashkov, D.Svyatskiy

Funding: DOE Office of Science, ASCR ProgramAcknowledgements: R.Garimella, MSTK,

(software.lanl.gov/MeshTools/trac)

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

http://math.lanl.gov/

Objective

The mimetic finite difference method preserves or mimicscritical mathematical and physical properties of systems ofPDEs such as conservation laws, exact identities, solutionsymmetries, secondary equations, maximum principles, etc.

These properties are important for multiphysics simulations.

The task of building mimetic schemes becomes more difficulton unstructured polygonal and polyhedral meshes.


Outline

1 Discrete vector and tensor calculusCoordinate invariant definition of primary mimeticoperator

Duality & derived mimetic operators

Properties of mimetic operators

2 Mimetic inner productsConsistency condition

Stability condition

Numerical example

3 Flexibility of mimetic discretization frameworkNonlinear parabolic problem

M-adaptation

Selection of DOFs (meshes with curved faces; Stokes)


Mesh notation

n – node, discrete space Nh

e – edge, length |e|, tangent τ e, discrete space Ehf – face, area |f |, normal nf , discrete space Fh

c – cell, volume |c|, discrete space ChKonstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Engineering mesh


Discrete vector and tensor calculus

Coordinate invariant definition of primary mimeticoperators1

Duality & derived mimetic operators

Properties of mimetic operators

1K.L., M.Manzini, M.Shashkov, JCP 2014Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators

Primary mimetic operators appear naturally from the Stokestheorem in one, two and three dimensions.∫e

∂p

∂τ edx = p(xn2)− p(xn1)

(GRADh ph

)e

=pn2 − pn1

|e|

∫f(curl u)·nf dx =

∮∂f

u·τ dx(CURLh uh

)f

=1

|f |∑e∈∂f

αf,e |e|ue

∫cdivudx =

∮∂c

u · ndx(DIVh uh

)c

=1

|c|∑f∈∂c

αc,f |f |uf

where α = ±1 and degrees of freedom are

pn = p(xn), ue =1

|e|

∫eu · τ e dx, uf =

1

|f |

∫fu · nf dx




∂p


(GRADh ph

)e

=pn2 − pn1

|e|


∮∂f

u·τ dx(CURLh uh

)f

=1

|f |∑e∈∂f

αf,e |e|ue

∫cdivudx =

∮∂c

u · ndx(DIVh uh

)c

=1

|c|∑f∈∂c

αc,f |f |uf


pn = p(xn), ue =1

|e|


1

|f |

∫fu · nf dx




∂p


(GRADh ph

)e

=pn2 − pn1

|e|


∮∂f

u·τ dx(CURLh uh

)f

=1

|f |∑e∈∂f

αf,e |e|ue

∫cdivudx =

∮∂c

u · ndx(DIVh uh

)c

=1

|c|∑f∈∂c

αc,f |f |uf


pn = p(xn), ue =1

|e|


1

|f |

∫fu · nf dx


Duality & derived mimetic operators (1/3)

The integration by part formula is∫Ω

(divu) q dx = −∫

Ωu · ∇q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0 (Ω)

In other words, ∇ = −div∗ with respect to L2 products.

We

define GRADh = −DIV∗h with respect to inner products[DIVhuh, qh

]Ch

= −[uh, GRADhqh

]Fh

∀uh ∈ Fh, qh ∈ Ch

The primary and derived mimetic operators (rectangularmatrices) are not discretized independently of one another.



The integration by part formula is∫Ω

(divu) q dx = −∫

Ωu · ∇q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0 (Ω)

In other words, ∇ = −div∗ with respect to L2 products. We

define GRADh = −DIV∗h with respect to inner products[DIVhuh, qh

]Ch

= −[uh, GRADhqh

]Fh


The primary and derived mimetic operators (rectangularmatrices) are not discretized independently of one another.



An inner product is defined by an SPD matrix MQ:

[uh, vh]Qh= (uh)T MQ vh, ∀uh,vh ∈ Qh.

Using this in the discrete duality formula, we have

GRADh = −M−1F (DIVh)T MC



Similarly to the derived gradient operator, we have

CURLh = M−1E (CURLh)T MF

andDIVh = −M−1

N (GRADh)T ME

Derived mimetic operators are fully characterized by theinner products and primary mimetic operators.


Discrete Laplacians (1/2)

The first discrete Laplacian is

∆h = DIVh GRADh : Ch → Ch

Using the definition of the derived gradient operator:

∆h = −DIVhM−1F (DIVh)T MC

Hence, we have symmetry and definiteness:

[∆h qh, ph]Ch = −qThMC DIVhM−1F (DIVh)T MC ph = [∆h ph, qh]Ch

and

[∆h qh, qh]Ch = −∥∥∥M−1/2F (DIVh)T MC qh

∥∥∥2≤ 0



The first discrete Laplacian is

∆h = DIVh GRADh : Ch → Ch

Using the definition of the derived gradient operator:

∆h = −DIVhM−1F (DIVh)T MC


[∆h qh, ph]Ch = −qThMC DIVhM−1F (DIVh)T MC ph = [∆h ph, qh]Ch

and

[∆h qh, qh]Ch = −∥∥∥M−1/2F (DIVh)T MC qh

∥∥∥2≤ 0



The second discrete Laplacian is

∆h = DIVh GRADh : Nh → Nh

Using the definition of the derived divergence operator:

∆h = −M−1N (GRADh)T ME GRADh


[∆h qh, ph]Nh= −qTh (GRADh)T ME GRADh ph = [∆h ph, qh]Nh

and

[∆h qh, qh]Nh= −

∥∥∥M1/2E GRADh qh

∥∥∥2≤ 0


Exact identities

By construction, we have the exact identities:

DIVh CURLh = 0 and CURLh GRADh = 0.

The derived operators satisfy similar identities:

DIVh CURLh = −(M−1N (GRADh)T ME

)(M−1E (CURLh)T MF

)= −M−1

N (CURLh GRADh)TMF = 0

and

CURLh GRADh = −(M−1E (CURLh)T MF

)(M−1F (DIVh)T MC

)= −M−1

E (DIVh CURLh)T MC = 0.


Exact identities

By construction, we have the exact identities:

DIVh CURLh = 0 and CURLh GRADh = 0.

The derived operators satisfy similar identities:

DIVh CURLh = −(M−1N (GRADh)T ME

)(M−1E (CURLh)T MF

)= −M−1

N (CURLh GRADh)TMF = 0

and

CURLh GRADh = −(M−1E (CURLh)T MF

)(M−1F (DIVh)T MC

)= −M−1

E (DIVh CURLh)T MC = 0.


Helmholtz decomposition theorems

Theorem I

Let domain Ω and mesh Ωh be simply-connected. Then, forany vh ∈ Fh there exists a unique qh ∈ Ch and a unique uh ∈ Ehwith DIVh uh = 0 such that

vh = GRADh qh + CURLh uh

Theorem II

Let domain Ω and mesh Ωh be simply-connected. Then, forany vh ∈ Eh there exist a discrete field qh ∈ Nh, which isdefined up to a constant field, and a unique discrete fielduh ∈ Fh with DIVh uh = 0 such that

vh = GRADh qh + CURLh uh


Related methods

Incomplete list of various compatible discretization methodsand frameworks includes

Cell method

Compatible discrete operators

Co-volume method

Summation by parts

Hybrid FV, mixed FV, discrete duality FV

Mixed FE, weak Galerkin, VEM, Kuznetsov-Repin

Exterior calculus


Special properties the mimetic framework

There is a lot of freedom in construction of primary andderived operators. This is especially improtant for PDEs withnon-constant coefficients. Using the weighed L2 product,∫

Ω(divu) q dx = −

∫Ωk−1u · (k∇)q dx,

we construct primary DIVh that approximates div(·) and

derived GRADh that approximates k∇(·).

Using∫Ω

(div (k u)) q dx = −∫

Ωku · ∇ q dx,

we construct primary DIVh that approximates div(k ·) and

derived GRADh that approximates ∇(·).


Special properties the mimetic framework

There is a lot of freedom in construction of primary andderived operators. This is especially improtant for PDEs withnon-constant coefficients. Using the weighed L2 product,∫

Ω(divu) q dx = −

∫Ωk−1u · (k∇)q dx,

we construct primary DIVh that approximates div(·) and

derived GRADh that approximates k∇(·). Using∫Ω

(div (k u)) q dx = −∫

Ωku · ∇ q dx,

we construct primary DIVh that approximates div(k ·) and

derived GRADh that approximates ∇(·).


Energy conservation (1/2)

The equation of Lagrangian gasdynamic (density ρ, velocityu, internal energy e, pressure p):

1

ρ

dρ

dt= −divu

ρdu

dt= −∇ p

ρde

dt= −p divu

Let p = 0 of ∂Ω. The integration by parts and continuityequation lead to conservation of the total energy E:

dE

dt=

∫Ω(t)

ρ(dudt·u+

de

dt

)dx = −

∫Ω(t)

(u ·∇p+p divu) dx = 0.


Energy conservation (2/2)

The semi-discrete equations read

1

ρh

dρhdt

= −DIVhuh

ρhduh

dt= −GRADh ph

ρhdehdt

= −phDIVhuh

The discrete integration by parts formula guaranteesconservation of the total discrete energy Eh:

dEh

dt= −[uh, GRADh ph]Fh

− [ph, DIVhuh]Ch = 0.


Divergence free discrete fields (1/2)

Maxwell’s equations (magnetic field H = µB, magnetic fluxdensity B, dielectric displacement D = εE, electric field E):

∂B

∂t= −curlE, ∂D

∂t= curlH,

satisfydivB = 0, divD = 0

for any time t.


Divergence free discrete fields (2/2)

The semi-discrete equations read

∂Bh

∂t= −CURLhEh,

∂Dh

∂t= CURLhHh

The exact discrete identities guarantee that the initialdivergence-free condition is preserved:

∂

∂t

(DIVhBh

)= DIVh

∂Bh

∂t= −DIVh CURLhEh = 0

and

∂

∂t

(DIVhDh

)= DIVh

∂Dh

∂t= −DIVh CURLhHh = 0.


Mimetic inner products

Consistency condition2

Stability condition

Numerical example

2F.Brezzi, K.L., V.Simoncini, M3AS 2005Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Recall formulas for derived operators

GRADh = −M−1F (DIVh)T MC

CURLh = M−1E (CURLh)T MF

DIVh = −M−1N (GRADh)T ME


Local inner products

Inner products are built cell-by-cell:[vh, uh]Fh

=∑c∈Ωh

[vc,h, uc,h]c,Fh

The cell-based inner product is defined by SPD matrix Mc,F :

[vc,h, uc,h]c,Fh

= (vc,h)TMc,F uc,h ≈∫cv · udx

Derivation of an accurate inner product is based on theconsistency and stability conditions. The inner productmatrix Mc,F is not unique.


Consistency condition (1/3)

First, we replace u with its best constant approximation u0:[vc,h, u

0c,h]c,Fh

≈∫cv · u0 dx

For any u0 there exists a linear polynomial q1 such that

u0 = ∇q1 and

∫cq1dx = 0.



Second, we integrate by parts:[vc,h, u

0c,h]c,Fh

= (vc,h)TMc,F u0c,h ≈

∫cv · ∇q1 dx

= −∫cq1divv dx+

∫∂cq1 v · ndx

≈ −DIVcvc,h

∫cq1 dx+

∑f∈∂c

vf

∫fq1 dx

Third, we set

(vc,h)TMc,F u0c,h =

∑f∈∂c

vf

∫fq1 dx ∀vc,h

Since vc,h is arbitrary, we conclude that Mc,F u0c,h = rc,h.



Second, we integrate by parts:[vc,h, u

0c,h]c,Fh

= (vc,h)TMc,F u0c,h ≈

∫cv · ∇q1 dx

= −∫cq1divv dx+

∫∂cq1 v · ndx

≈ −DIVcvc,h

∫cq1 dx+

∑f∈∂c

vf

∫fq1 dx

Third, we set

(vc,h)TMc,F u0c,h =

∑f∈∂c

vf

∫fq1 dx ∀vc,h

Since vc,h is arbitrary, we conclude that Mc,F u0c,h = rc,h.



Algebraic equations w.r.t. unknown matrix Mc,F :

Mc,F

u0f1...

u0fm

=

∫f1

q1 dx

...∫fm

q1 dx

∀u0 = ∇q1

It is sufficient to consider only linearly independent functionsq1. In 3D, we have q1

a = x− xc, q1b = y − yc, and q1

c = z − zc.

Mc,F︸︷︷︸m×m

Nc︸︷︷︸m×3

= Rc︸︷︷︸m×3

.

The problem is under-determined for any cell c (triangles:Shashkov, Hyman; Shashkov, Liska; Nicolaides, Trapp).


Construction of Nc and Rc for a hexahedron

Mc,F Nc = Rc

Required geometric information: normals to faces, centroidsof faces, areas of faces, centroid of the cell:

Nc =

n1x n1y n1z

n2x n2y n2z

......

...

n6x n6y n6z

Rc =

|f1|(x1 − xc) |f1|(y1 − yc) |f1|(z1 − zc)

|f2|(x1 − xc) |f2|(y2 − yc) |f2|(z2 − zc)

......

...

|f6|(x6 − xc) |f6|(y6 − yc) |f6|(z6 − zc)


Properties of Nc and Rc

Lemma

For any polyhedron, we have

NTc Rc = RT

c Nc = |c| I.


Solution of the mimetic matrix equation

Lemma

A family of SPD solutions to Mc,FNc = Rc is

Mc,F = Mconsistencyc,F + Mstability

c,F

where

Mconsistencyc,F =

1

|c|RcRT

c

and

Mstabilityc,F =

(I− Nc

(NTc Nc

)−1 NTc

)Pc

(I− Nc

(NTc Nc

)−1 NTc

)where Pc is an SPD matrix.


Stability condition (1/2)

Consider a model elliptic problem and calculate Darcy fluxand pressure errors as functions of one normalize parameter.

Pc = acI

The free parameter acmay vary 2-orders inmagnitude.


Stability condition (2/2)

Mc,F should behave like a mass matrix:

σ?|c| ‖vc,h‖2 ≤[vc,h, vc,h]c,Fh

≤ σ?|c| ‖vc,h‖2

where σ? and σ? are independent of mesh. This imposesrestriction on the parameter matrix:

σ?|c| ‖vc,h‖2 ≤ vTc,hM

consistencyc,F vc,h+vT

c,hMstabilityc,F vc,h ≤ σ?|c| ‖vc,h‖2

In practice, a good choice is given by the scalar matrix

Pc =1

3|c| I.


Connection with VEM

Consider the following infinite-dimensional space

Sc =v : v · nf ∈ P 0(f), divv ∈ P 0(c)

The consistency condition is the exactness property:[

vc,h, u0c,h]c,Fh

=

∫cv · u0 dx ∀u0 ∈ P 0(c), ∀v ∈ Sc.

Restricting further the space Sc, we get a VEM space.


Darcy problem: Convergence results

uh = −GRADh ph DIVh uh = bIh.

Let Ω have a Lipschitz continuous boundary;

every cell c be shape regular;

pIh ∈ Ch, uIh ∈ Fh be interpolants of exact solution. Then

|||pIh − ph|||Ch + |||uIh − uh|||Fh

≤ C h

where h is the mesh parameter.3, If Ω is convex and λmin(Pc)is sufficiently large, then

|||pIh − ph|||Ch ≤ C h2

Framework of gradient schemes can be also used for theconvergence analysis.4

3F.Brezzi, K.L., M.Shashkov; SINUM 20054J.Droniou, R.Eymard, T.Gallouet, R.Herbin, M3AS, 2013


Non-matching randomly perturbed meshes

K1 = 1,K2 = 106

aspect ratio variations:

167 < maxcells

maxk |fk|mink |fk|

< 2024exact solution is

p(x, y) =

7

16−

K2

12K1

+2K2

3K1

y3, y < 0.5,

y − y4, y ≥ 0.5.


Flexibility of the MFD framework

Nonlinear parabolic problem5

M-adaptation


5K.L., M.Manzini, J.Moulton, M.Shashkov, JCP 2015Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Harmonic averaging vs arithmetic averaging

∂p

∂t− ∂

∂x(k(p)

∂p

∂x) = 0, k(p) = p3

Initial condition p(x, 0) = 10−3, left b.c. is p(0, t) = 1.44 3√t.

1 Harmonic averaging (left): uf = − 2kL kRkL + kR

pR − pLh

→ 0

as kR → 0.

2 Arithmetic averaging (right): uf = −kL + kR2

pR − pLh

leads to the correct solution.Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Application I: heat transfer

∂(ρ cv T )

∂t− div(k(T )∇T ) = 0, k(T ) = T 3

Consider the above problem in 2D and increase the initialcondition: T (x, 0) = 0.02. MFE, VEM, old MFD, and manyothers are effectively methods with harmonic averaging:

The heat wave is moving slightly faster than in 1D example,but still too slow.


Application II: infiltration in a dry soil

∂θ(p)

∂t− div

(k(p)(∇p− ρg)

)= 0

where θ is water content, k(p) is highly nonlinear function.6

6ASCEM, software.lanl.gov/ascem/amanziKonstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Application II: infiltration in a dry soil

∂θ(p)

∂t− div

(k(p)(∇p− ρg)

)= 0

where θ is water content, k(p) is highly nonlinear function.6

6ASCEM, software.lanl.gov/ascem/amanziKonstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Known solutions and their limitations

Refine mesh around the moving front. Should work forRichards’s equation but breaks down for other physicalmodels that allow k(p) = 0 (e.g. surface water7).

Two-velocity formulation (enhanced MFE8).

u = −∇p+ ρg

v = k(p)u∂θ

∂t+ divv = 0

The discrete system is symmetric only for the case ofcell-centered diffusion coefficients.

7E.Coon, J.Moulton, M.Berndt, G.Manzini, R.Garimella, K.L., S.Painter,AWR 2015

8T.Arbogast, C.Dawson, P.Keenan, M.Wheeler, I.Yotov; SISC, 1998Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Primary and dual mimetic operators (1/2)

∂p

∂t+ div

(k u)

= 0, u = −∇p.

Consider the integration by parts formula:∫Ω

(div (k u)) q dx = −∫

Ωk u · ∇ q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0 (Ω)

In other words, ∇(·) = −(divk(·))∗ with respect to theweighed inner products.

We define DIVh as approximation of

divk(·) and GRADh = −DIV∗h with respect to inner product[DIVhuh, qh

]Ch

= −[uh, GRADhqh

]Fh




∂p

∂t+ div

(k u)

= 0, u = −∇p.

Consider the integration by parts formula:∫Ω

(div (k u)) q dx = −∫

Ωk u · ∇ q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0 (Ω)

In other words, ∇(·) = −(divk(·))∗ with respect to theweighed inner products. We define DIVh as approximation of

divk(·) and GRADh = −DIV∗h with respect to inner product[DIVhuh, qh

]Ch

= −[uh, GRADhqh

]Fh




Using a variation of the Stokes formula,∫cdiv (k u) dx =

∮∂ck u · n dx,

we define the primary mimetic operator as(DIVh uh

)c

=1

|c|∑f∈∂c

αc,f kf |f |uf

We may use different models to define kf on mesh faces:harmonic averaging, arithmetic averaging, upwinding, etc.

The derived mimetic operator has the typical structure:

GRADh = −M−1F (DIVh)T MC .


Inner product

The inner product in the space of discrete gradients,[vh, uh

]c,Fh≈∫ck v · udx ≈ kc

∫cv · udx,

can be derived using the above arguments.


MFD for nonlinear parabolic equations (2/2)

∂(ρ cv T )

∂t− div(k(T )∇T ) = 0, k(T ) = T 3.

The initial condition T (x, 0) = 0.02. The left b.c.T (0, t) = 0.78 3

√t results in a wave moving from left to right:

In the new MFD scheme this wave moves with the correctspeed.



Nonlinear parabolic problem

M-adaptation910


9V.Gyrya, K.L., JCA 201210K.L., Proceedings of FVCA14


How rich is the family of MFD schemes?

Mc,F = Mconsistencyc,F +

(I− Nc

(NTc Nc

)−1 NTc

)Pc

(I− Nc

(NTc Nc

)−1 NTc

)Cell # parameters

triangle/tetrahedron 1

quadrilateral 3

hexahedron 6

tetradecahedron 66


Acoustic wave equation

Analysis of the family of mimetic schemes lead to discoveryof a new scheme with the six-order numerical anisotropy.


Monotone mimetic schemes: error reduction

k =

[(x+ 1)2 + y2 −xy−xy (x+ 1)2

], p = x3y2 + x sin(2πx) sin(2πy)

Two mesh-generators were used.11

11Ani2D (sourceforge.net/projects/ani2d/) and MSTK ToolSet.Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Monotone mimetic schemes: solution positivity (1/2)

∂(φC)

∂t+div(uC) = −div(k∇C), k = αL

uu

‖u‖2+αT

(I− uu

‖u‖2)

velocity u makes angle 30 with the mesh orientation.

There exists a monotone scheme!


Monotone mimetic schemes: solution positivity (2/2)

Non-optimized MFD scheme leads to C < 0. Even smalloscillations may be amplified by chemical reactions12.

12C.Steefel, K.MacQuarrie, Reviews in Mineralogy 1996Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes


Nonlinear parabolic problem

M-adaptation

Selection of DOFs ( meshes with curved faces13;Stokes14)

13F.Brezzi, K.L., M.Shashkov, V.Simoncini, CMAME 200714L.Beirao da Veiga, K.L., SISC 2010


Hexahedral meshes with curved faces

Methods with one velocity unknown per curved mesh face donot converge. MFD technology allows to use 3 unknowns.


Stokes: Stabilizing DOFs (1/3)

No bubble DOFs are needed

1/h β ε0(u) ε1(u) ε0(p)

8 2.05e-1 2.09e-1 2.31e-1 4.14e-016 2.02e-1 6.47e-2 1.01e-1 1.16e-032 2.00e-1 1.73e-2 4.55e-2 3.01e-164 2.00e-1 4.42e-3 2.20e-2 7.75e-2

128 1.99e-1 1.11e-3 1.09e-2 2.01e-2



52% of edges have bubble DOFs

1/h β ε0(u) ε1(u) ε0(p)


128 1.30e-1 7.22e-4 1.53e-3 4.96e-2



25% of edges have bubble DOFs

1/h β ε0(u) ε1(u) ε0(p)


128 9.53e-2 4.75e-4 8.63e-3 9.23e-2


Conclusion

The mimetic finite difference method is designed tomimic important properties of mathematical and physicalsystems on arbitrary polygonal or polyhedral meshes.

The MFD method for diffusion problems is relative easyto implement on general polyhedral meshes. Same istrue for other PDEs.

The flexibility of the MFD framework has been used todevelop new schemes for nonlinear parabolic equationswith small diffusion coefficients; optimize mimeticschemes; and add DOFs as needed for accuracy orstability.


Design Principles of the Mimetic Finite Difference Schemes · 2015-10-28 · Design Principles of the Mimetic Finite Di erence Schemes Konstantin Lipnikov Los Alamos National Laboratory,

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