design principles of the mimetic finite difference schemessimilarly to the derived gradient...
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![Page 1: Design Principles of the Mimetic Finite Difference SchemesSimilarly to the derived gradient operator, we have CURL^ h = M 1 E (CURL h) T M F and DIVg h = M 1 N (GRAD h) T M E Derived](https://reader036.vdocuments.site/reader036/viewer/2022071511/61307ed21ecc515869442263/html5/thumbnails/1.jpg)
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
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
![Page 3: Design Principles of the Mimetic Finite Difference SchemesSimilarly to the derived gradient operator, we have CURL^ h = M 1 E (CURL h) T M F and DIVg h = M 1 N (GRAD h) T M E Derived](https://reader036.vdocuments.site/reader036/viewer/2022071511/61307ed21ecc515869442263/html5/thumbnails/3.jpg)
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)
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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Engineering mesh
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
![Page 9: Design Principles of the Mimetic Finite Difference SchemesSimilarly to the derived gradient operator, we have CURL^ h = M 1 E (CURL h) T M F and DIVg h = M 1 N (GRAD h) T M E Derived](https://reader036.vdocuments.site/reader036/viewer/2022071511/61307ed21ecc515869442263/html5/thumbnails/9.jpg)
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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Duality & derived mimetic operators (2/3)
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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Duality & derived mimetic operators (3/3)
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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Discrete Laplacians (2/2)
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
Hence, we have symmetry and definiteness:
[∆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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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 ∇(·).
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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 ∇(·).
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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Recall formulas for derived operators
GRADh = −M−1F (DIVh)T MC
CURLh = M−1E (CURLh)T MF
DIVh = −M−1N (GRADh)T ME
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Consistency condition (2/3)
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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Consistency condition (2/3)
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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Consistency condition (3/3)
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).
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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)
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Properties of Nc and Rc
Lemma
For any polyhedron, we have
NTc Rc = RT
c Nc = |c| I.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Flexibility of the MFD framework
Nonlinear parabolic problem5
M-adaptation
Selection of DOFs (meshes with curved faces; Stokes)
5K.L., M.Manzini, J.Moulton, M.Shashkov, JCP 2015Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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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
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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
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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
∀uh ∈ Fh, qh ∈ Ch
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
∀uh ∈ Fh, qh ∈ Ch
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Primary and dual mimetic operators (2/2)
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 .
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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Flexibility of the MFD framework
Nonlinear parabolic problem
M-adaptation910
Selection of DOFs (meshes with curved faces; Stokes)
9V.Gyrya, K.L., JCA 201210K.L., Proceedings of FVCA14
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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Acoustic wave equation
Analysis of the family of mimetic schemes lead to discoveryof a new scheme with the six-order numerical anisotropy.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
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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
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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!
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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
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Flexibility of the MFD framework
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
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Hexahedral meshes with curved faces
Methods with one velocity unknown per curved mesh face donot converge. MFD technology allows to use 3 unknowns.
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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
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Stokes: Stabilizing DOFs (2/3)
52% of edges have bubble DOFs
1/h β ε0(u) ε1(u) ε0(p)
8 1.29e-1 1.57e-1 1.24e-1 3.55e-016 1.30e-1 4.35e-2 4.41e-2 1.20e-032 1.29e-1 1.13e-2 1.46e-2 4.25e-164 1.32e-1 2.86e-3 4.71e-3 1.45e-1
128 1.30e-1 7.22e-4 1.53e-3 4.96e-2
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Stokes: Stabilizing DOFs (3/3)
25% of edges have bubble DOFs
1/h β ε0(u) ε1(u) ε0(p)
8 9.15e-2 1.43e-1 2.17e-1 4.24e-016 1.16e-1 3.28e-2 9.24e-2 1.53e-032 6.58e-2 7.94e-3 4.34e-2 6.22e-164 6.63e-2 1.84e-3 1.84e-2 2.12e-1
128 9.53e-2 4.75e-4 8.63e-3 9.23e-2
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes
![Page 64: Design Principles of the Mimetic Finite Difference SchemesSimilarly to the derived gradient operator, we have CURL^ h = M 1 E (CURL h) T M F and DIVg h = M 1 N (GRAD h) T M E Derived](https://reader036.vdocuments.site/reader036/viewer/2022071511/61307ed21ecc515869442263/html5/thumbnails/64.jpg)
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.
Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes