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Design Principles of the Mimetic Finite Difference Schemes

Konstantin Lipnikov

Los Alamos National Laboratory, Theoretical Division Applied 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 Program Acknowledgements: R.Garimella, MSTK, (software.lanl.gov/MeshTools/trac)

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Objective

The mimetic finite difference method preserves or mimics critical mathematical and physical properties of systems of PDEs such as conservation laws, exact identities, solution symmetries, secondary equations, maximum principles, etc. These properties are important for multiphysics simulations. The task of building mimetic schemes becomes more difficult

• n unstructured polygonal and polyhedral meshes.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Outline

1 Discrete vector and tensor calculus

Coordinate invariant definition of primary mimetic

• perator

Duality & derived mimetic operators Properties of mimetic operators

2 Mimetic inner products

Consistency condition Stability condition Numerical example

3 Flexibility of mimetic discretization framework

Nonlinear parabolic problem M-adaptation Selection of DOFs (meshes with curved faces; Stokes)

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Mesh notation

n – node, discrete space Nh e – edge, length |e|, tangent τ e, discrete space Eh f – face, area |f|, normal nf, discrete space Fh c – cell, volume |c|, discrete space Ch

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Engineering mesh

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete vector and tensor calculus

Coordinate invariant definition of primary mimetic

• perators1

Duality & derived mimetic operators Properties of mimetic operators

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

Coordinate invariant definition of primary operators

Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions.

• e

∂p ∂τ e dx = p(xn2) − p(xn1)

• GRADh ph
• e = pn2 − pn1

|e|

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators

Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions.

• e

∂p ∂τ e dx = 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

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators

Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions.

• e

∂p ∂τ e dx = 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

• c

div u dx =

• ∂c

u · n dx

• 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|

• e

u · τ e dx, uf = 1 |f|

• f

u · nf dx

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (1/3)

The integration by part formula is

• Ω

(div u) q dx = −

• Ω

u · ∇q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0(Ω)

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

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (1/3)

The integration by part formula is

• Ω

(div u) q dx = −

• Ω

u · ∇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 (rectangular matrices) are not discretized independently of one another.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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−1

F (DIVh)T MC

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (3/3)

Similarly to the derived gradient operator, we have

• CURLh = M−1

E (CURLh)T MF

and

• DIVh = −M−1

N (GRADh)T ME

Derived mimetic operators are fully characterized by the inner products and primary mimetic operators.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (1/2)

The first discrete Laplacian is ∆h = DIVh GRADh : Ch → Ch Using the definition of the derived gradient operator: ∆h = −DIVh M−1

F (DIVh)T MC

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (1/2)

The first discrete Laplacian is ∆h = DIVh GRADh : Ch → Ch Using the definition of the derived gradient operator: ∆h = −DIVh M−1

F (DIVh)T MC

Hence, we have symmetry and definiteness: [∆h qh, ph]Ch = −qT

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

and [∆h qh, qh]Ch = −

• M−1/2

F

(DIVh)T MC qh

• 2

≤ 0

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (2/2)

The second discrete Laplacian is ∆h = DIVh GRADh : Nh → Nh Using the definition of the derived divergence operator: ∆h = −M−1

N (GRADh)T ME GRADh

Hence, we have symmetry and definiteness: [∆h qh, ph]Nh = −qT

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

and [∆h qh, qh]Nh = −

• M1/2

E

GRADh qh

• 2

≤ 0

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Exact identities

By construction, we have the exact identities: DIVh CURLh = 0 and CURLh GRADh = 0.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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−1

N (GRADh)T ME

• M−1

E (CURLh)T MF

• =

−M−1

N (CURLh GRADh)T MF = 0

and

• CURLh

GRADh = −

• M−1

E (CURLh)T MF

• M−1

F (DIVh)T MC

• =

−M−1

E (DIVh CURLh)T MC = 0.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Helmholtz decomposition theorems

Theorem I Let domain Ω and mesh Ωh be simply-connected. Then, for any vh ∈ Fh there exists a unique qh ∈ Ch and a unique uh ∈ Eh with DIVh uh = 0 such that vh = GRADh qh + CURLh uh Theorem II Let domain Ω and mesh Ωh be simply-connected. Then, for any vh ∈ Eh there exist a discrete field qh ∈ Nh, which is defined up to a constant field, and a unique discrete field uh ∈ Fh with DIVh uh = 0 such that vh = GRADh qh + CURLh uh

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Related methods

Incomplete list of various compatible discretization methods and 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

Special properties the mimetic framework

There is a lot of freedom in construction of primary and derived operators. This is especially improtant for PDEs with non-constant coefficients. Using the weighed L2 product,

• Ω

(div u) q dx = −

• Ω

k−1u · (k ∇)q dx, we construct primary DIVh that approximates div(·) and derived GRADh that approximates k ∇(·).

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Special properties the mimetic framework

There is a lot of freedom in construction of primary and derived operators. This is especially improtant for PDEs with non-constant coefficients. Using the weighed L2 product,

• Ω

(div u) q dx = −

• Ω

k−1u · (k ∇)q dx, we construct primary DIVh that approximates div(·) and derived GRADh that approximates k ∇(·). Using

• Ω

(div (k u)) q dx = −

• Ω

ku · ∇ 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

Energy conservation (1/2)

The equation of Lagrangian gasdynamic (density ρ, velocity u, internal energy e, pressure p): 1 ρ dρ dt = −div u ρdu dt = −∇ p ρde dt = −p div u Let p = 0 of ∂Ω. The integration by parts and continuity equation lead to conservation of the total energy E: dE dt =

• Ω(t)

ρ du dt ·u+ de dt

• dx = −
• Ω(t)
• u·∇p+p div u) dx = 0.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Energy conservation (2/2)

The semi-discrete equations read 1 ρh dρh dt = −DIVhuh ρh duh dt = − GRADh ph ρh deh dt = −ph DIVhuh The discrete integration by parts formula guarantees conservation 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

Divergence free discrete fields (1/2)

Maxwell’s equations (magnetic field H = µB, magnetic flux density B, dielectric displacement D = ǫE, electric field E): ∂B ∂t = −curl E, ∂D ∂t = curl H, satisfy divB = 0, divD = 0 for any time t.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Divergence free discrete fields (2/2)

The semi-discrete equations read ∂Bh ∂t = −CURLh Eh, ∂Dh ∂t = CURLh Hh The exact discrete identities guarantee that the initial divergence-free condition is preserved: ∂ ∂t

• DIVh Bh
• = DIVh

∂Bh ∂t = −DIVh CURLh Eh = 0 and ∂ ∂t DIVh Dh

• =

DIVh ∂Dh ∂t = − DIVh CURLh Hh = 0.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Mimetic inner products

Consistency condition2 Stability condition Numerical example

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

Recall formulas for derived operators

• GRADh = −M−1

F (DIVh)T MC

• CURLh = M−1

E (CURLh)T MF

• DIVh = −M−1

N (GRADh)T ME

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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)T Mc,F uc,h ≈
• c

v · u dx Derivation of an accurate inner product is based on the consistency and stability conditions. The inner product matrix Mc,F is not unique.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Consistency condition (1/3)

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

• vc,h, u0

c,h]c,Fh ≈

• c

v · u0 dx For any u0 there exists a linear polynomial q1 such that u0 = ∇q1 and

• c

q1dx = 0.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Consistency condition (2/3)

Second, we integrate by parts:

• vc,h, u0

c,h]c,Fh = (vc,h)T Mc,F u0 c,h ≈

• c

v · ∇q1 dx = −

• c

q1divv dx +

• ∂c

q1 v · n dx ≈ −DIVcvc,h

• c

q1 dx +

• f∈∂c

vf

• f

q1 dx

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Consistency condition (2/3)

Second, we integrate by parts:

• vc,h, u0

c,h]c,Fh = (vc,h)T Mc,F u0 c,h ≈

• c

v · ∇q1 dx = −

• c

q1divv dx +

• ∂c

q1 v · n dx ≈ −DIVcvc,h

• c

q1 dx +

• f∈∂c

vf

• f

q1 dx Third, we set (vc,h)T Mc,F u0

c,h =

• f∈∂c

vf

• f

q1 dx ∀vc,h Since vc,h is arbitrary, we conclude that Mc,F u0

c,h = rc,h.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Consistency condition (3/3)

Algebraic equations w.r.t. unknown matrix Mc,F: Mc,F    u0

f1

. . . u0

fm

   =       

• f1

q1 dx . . .

• fm

q1 dx        ∀u0 = ∇q1 It is sufficient to consider only linearly independent functions

• q1. In 3D, we have q1

a = x − xc, q1 b = 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

Construction of Nc and Rc for a hexahedron

Mc,F Nc = Rc Required geometric information: normals to faces, centroids

• f 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

Properties of Nc and Rc

Lemma For any polyhedron, we have NT

c Rc = RT c Nc = |c| I.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Solution of the mimetic matrix equation

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

c,F

+ Mstability

c,F

where Mconsistency

c,F

= 1 |c|Rc RT

c

and Mstability

c,F

=

• I − Nc
• NT

c Nc

−1 NT

c

• Pc
• I − Nc
• NT

c Nc

−1 NT

c

• where Pc is an SPD matrix.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Stability condition (1/2)

Consider a model elliptic problem and calculate Darcy flux and pressure errors as functions of one normalize parameter. Pc = acI The free parameter ac may vary 2-orders in magnitude.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Stability condition (2/2)

Mc,F should behave like a mass matrix: σ⋆|c| vc,h2 ≤

• vc,h, vc,h]c,Fh ≤ σ⋆|c| vc,h2

where σ⋆ and σ⋆ are independent of mesh. This imposes restriction on the parameter matrix: σ⋆|c| vc,h2 ≤ vT

c,h Mconsistency c,F

vc,h+vT

c,h Mstability c,F

vc,h ≤ σ⋆|c| vc,h2 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

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, u0

c,h]c,Fh =

• c

v · 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

Darcy problem: Convergence results

uh = − GRADh ph DIVh uh = bI

h.

Let Ω have a Lipschitz continuous boundary; every cell c be shape regular; pI

h ∈ Ch, uI h ∈ Fh be interpolants of exact solution. Then

|||pI

h − ph|||Ch + |||uI h − uh|||Fh ≤ C h

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

h − ph|||Ch ≤ C h2

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

3F.Brezzi, K.L., M.Shashkov; SINUM 2005 4J.Droniou, R.Eymard, T.Gallou¨

et, R.Herbin, M3AS, 2013

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Non-matching randomly perturbed meshes

K1 = 1, K2 = 106 aspect ratio variations: 167 < max

cells

maxk |fk| mink |fk| < 2024 exact 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

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 2015 Konstantin 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 kR

kL + kR pR − pL h → 0 as kR → 0.

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

2 pR − pL h 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 initial condition: T(x, 0) = 0.02. MFE, VEM, old MFD, and many

• thers 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

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/amanzi Konstantin 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/amanzi Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Known solutions and their limitations

Refine mesh around the moving front. Should work for Richards’s equation but breaks down for other physical models that allow k(p) = 0 (e.g. surface water7). Two-velocity formulation (enhanced MFE8). u = −∇p + ρg v = k(p) u ∂θ ∂t + div v = The discrete system is symmetric only for the case of cell-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, 1998 Konstantin 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 = −

• Ω

k u · ∇ q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0(Ω)

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

Konstantin 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 = −

• Ω

k u · ∇ q dx ∀u ∈ Hdiv(Ω), q ∈ H1

0(Ω)

In other words, ∇(·) = −(divk(·))∗ with respect to the weighed 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

Primary and dual mimetic operators (2/2)

Using a variation of the Stokes formula,

• c

div (k u) dx =

• ∂c

k 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−1

F (DIVh)T MC.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Inner product

The inner product in the space of discrete gradients,

• vh, uh
• c,Fh ≈
• c

k v · u dx ≈ kc

• c

v · u dx, can be derived using the above arguments.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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 correct speed.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Flexibility of the MFD framework

Nonlinear parabolic problem M-adaptation910 Selection of DOFs (meshes with curved faces; Stokes)

9V.Gyrya, K.L., JCA 2012 10K.L., Proceedings of FVCA14 Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

How rich is the family of MFD schemes?

Mc,F = Mconsistency

c,F

+

• I − Nc
• NT

c Nc

−1 NT

c

• Pc
• I − Nc
• NT

c Nc

−1 NT

c

• Cell

# parameters triangle/tetrahedron 1 quadrilateral 3 hexahedron 6 tetradecahedron 66

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Acoustic wave equation

Analysis of the family of mimetic schemes lead to discovery

• f a new scheme with the six-order numerical anisotropy.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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 u2 +αT

• I− uu

u2

• velocity u makes angle 30◦ with the mesh orientation.

There exists a monotone scheme!

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Monotone mimetic schemes: solution positivity (2/2)

Non-optimized MFD scheme leads to C < 0. Even small

• scillations may be amplified by chemical reactions12.

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

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 2007 14L.Beirao da Veiga, K.L., SISC 2010 Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Hexahedral meshes with curved faces

Methods with one velocity unknown per curved mesh face do not converge. MFD technology allows to use 3 unknowns.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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-0 16 2.02e-1 6.47e-2 1.01e-1 1.16e-0 32 2.00e-1 1.73e-2 4.55e-2 3.01e-1 64 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

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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-0 16 1.30e-1 4.35e-2 4.41e-2 1.20e-0 32 1.29e-1 1.13e-2 1.46e-2 4.25e-1 64 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

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

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-0 16 1.16e-1 3.28e-2 9.24e-2 1.53e-0 32 6.58e-2 7.94e-3 4.34e-2 6.22e-1 64 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

Conclusion

The mimetic finite difference method is designed to mimic important properties of mathematical and physical systems on arbitrary polygonal or polyhedral meshes. The MFD method for diffusion problems is relative easy to implement on general polyhedral meshes. Same is true for other PDEs. The flexibility of the MFD framework has been used to develop new schemes for nonlinear parabolic equations with small diffusion coefficients; optimize mimetic schemes; and add DOFs as needed for accuracy or stability.

Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes