[PPT] - The Mimetic Finite Difference Method Gianmarco Manzini 1 Istituto di PowerPoint Presentation

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Outline

The Mimetic Finite Difference Method

Gianmarco Manzini1

Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) C.N.R., Pavia, Italy

FVCA5 - June 08-13, 2008 Aussois, France

Manzini, G. The Mimetic Finite Difference Method

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Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

SLIDE 3

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

SLIDE 4

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

“Mimetic (mathematics)”

From Wikipedia, the free encyclopedia

(i) The goal of numerical analysis is to approximate the continuum, so instead of solving a partial differential equation one aims in solve a discrete version of the continuum problem. (ii) A numerical method is called mimetic when it mimics (or imitates) some properties of the continuum vector calculus. An example: a mixed finite element method applied to Darcy flows strictly conserves the mass of the flowing fluid.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Some literature. . .

Mimetic schemes were first proposed in the early eighties: Samarskii-Tishkin-Favorskii-Shashkov, Operational Finite-Difference Schemes, Differential Equations, 1981; many papers were published after this one. . . Some recent joint work from Los Alamos-Pavia: Brezzi-Lipnikov-Shashkov,SINUM,2005 (a priori estimates) Brezzi-Lipnikov-Simoncini, M3AS, 2005 (a family of MFDs) . . . Extensions: Cangiani-M. CMAME, 2008 (post-processing) Beirao da Veiga-M., NME, 2008 (mesh adaptivity) . . .

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

. . . people and topics (in Pavia)

Main features: family of schemes based on mixed formulation; grids formed by elements of general shape (polygons, polyhedra); people currently working in Pavia: Beirao da Veiga, Boffi, Brezzi, Buffa, Cangiani, M.,

A. Russo, . . .

some topics under investigation: diffusion and convection-diffusion models a posteriori estimates and mesh adaptivity electromagnetism Stokes equations 2-D software implementation

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Linear diffusion in mixed form

Consider div (−K∇p) = b, Ω ⊂ I Rd, d = 2, 3 +boundary conditions Let − → F be the flux vector variable: − → F = −K∇p constitutive equation div − → F = b conservation equation (1) Model problem: solve (1) for p and − → F with suitable boundary conditions

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Formally

FIRST, (let {Th} be a family of partitions of Ω formed by polygonal elements, h being the mesh size); (i) degrees of freedom for − scalar fields − → discrete scalars, Qh; − vector fields − → discrete vectors, Xh; Qh and Xh are not functions, but vectors of numbers! (ii) “discrete” operators: − the discrete divergence DIVh : Xh → Qh; − the discrete flux (or gradient) Gh : Qh → Xh; satisfying a duality relationship (discrete Gauss-Green formula).

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Formally

FIRST, (let {Th} be a family of partitions of Ω formed by polygonal elements, h being the mesh size); (i) degrees of freedom for − scalar fields − → discrete scalars, Qh; − vector fields − → discrete vectors, Xh; Qh and Xh are not functions, but vectors of numbers! (ii) “discrete” operators: − the discrete divergence DIVh : Xh → Qh; − the discrete flux (or gradient) Gh : Qh → Xh; satisfying a duality relationship (discrete Gauss-Green formula).

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Formally

FIRST, (let {Th} be a family of partitions of Ω formed by polygonal elements, h being the mesh size); (i) degrees of freedom for − scalar fields − → discrete scalars, Qh; − vector fields − → discrete vectors, Xh; Qh and Xh are not functions, but vectors of numbers! (ii) “discrete” operators: − the discrete divergence DIVh : Xh → Qh; − the discrete flux (or gradient) Gh : Qh → Xh; satisfying a duality relationship (discrete Gauss-Green formula).

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Formally

THEN, mimic the continuous differential equations by using discrete

perators acting on the discrete scalar and flux unknowns

ph ∈ Qh and Fh ∈ Xh: constitutive equation: − → F = −K∇p − → Fh = Ghph conservation equation: div − → F = b − → DIVhFh = bI (where bI is a suitable interpolation of b in Qh)

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Qh, degrees of freedom for scalar fields

Ω, Th E qE

q ∈ Qh

means q = ˘ qE ¯

E∈Th

(equivalent to a piecewise constant function)

dim(Qh) = number of elements
f the mesh.
”interpolation” operator:

(pI)E = 1 |E| Z

E

p dV.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Qh, degrees of freedom for scalar fields

Ω, Th E qE

q ∈ Qh

means q = ˘ qE ¯

E∈Th

(equivalent to a piecewise constant function)

dim(Qh) = number of elements
f the mesh.
”interpolation” operator:

(pI)E = 1 |E| Z

E

p dV.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Qh, degrees of freedom for scalar fields

Ω, Th E qE

q ∈ Qh

means q = ˘ qE ¯

E∈Th

(equivalent to a piecewise constant function)

dim(Qh) = number of elements
f the mesh.
”interpolation” operator:

(pI)E = 1 |E| Z

E

p dV.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Xh, degrees of freedom for vector fields

Ω, Th E E′ e

G ∈ Xh

means G = ˘ Ge

E

¯ e is an edge of E

Ge

E + Ge E′ = 0

∀e ⊆ E ∩ E′ dim(Xh) = number of edges

f the mesh.
”interpolation” operator:

`− → F I´e

E = 1

|e| Z

e

− → n e

E · −

→ F dV

− → n e

E

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Xh, degrees of freedom for vector fields

Ω, Th E E′ e

G ∈ Xh

means G = ˘ Ge

E

¯ e is an edge of E

Ge

E + Ge E′ = 0

∀e ⊆ E ∩ E′ dim(Xh) = number of edges

f the mesh.
”interpolation” operator:

`− → F I´e

E = 1

|e| Z

e

− → n e

E · −

→ F dV

− → n e

E

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Xh, degrees of freedom for vector fields

Ω, Th E E′ e

G ∈ Xh

means G = ˘ Ge

E

¯ e is an edge of E

Ge

E + Ge E′ = 0

∀e ⊆ E ∩ E′ dim(Xh) = number of edges

f the mesh.
”interpolation” operator:

`− → F I´e

E = 1

|e| Z

e

− → n e

E · −

→ F dV

− → n e

E

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Discrete divergence operator DIVh : Xh → Qh

Let − → G be a true vector field; take the cell average of div − → G

ver E and use Gauss Theorem:

1 |E|

E

div − → G dV = 1 |E|

∂E

− → n E · − → G dS = 1 |E|

e∈∂E

|e| − → GIe

E

Let G ∈ Xh; the discrete divergence of G in Qh is defined element by element as the constant value

DIVhG)E = 1

|E|

e∈∂E

|e| Ge

E

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Discrete divergence operator DIVh : Xh → Qh

Let − → G be a true vector field; take the cell average of div − → G

ver E and use Gauss Theorem:

1 |E|

E

div − → G dV = 1 |E|

∂E

− → n E · − → G dS = 1 |E|

e∈∂E

|e| − → GIe

E

Let G ∈ Xh; the discrete divergence of G in Qh is defined element by element as the constant value

DIVhG)E = 1

|E|

e∈∂E

|e| Ge

E

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The mimetic conservation equation: DIVhFh = bI

Take the cell average of the conservation equation: div − → F = b − → 1 |E|

E

div − → F dV = 1 |E|

E

b dV = bI|E and define the flux Fh as the solution to

DIVhFh)E = bI|E ≡ 1

|E|

E

div − → F dV using the inner product

p, q
Qh :=

E |E| pEqE =

Ω

p q dV, variational form:

DIVhFh, q
Qh =
bI, q
Qh

∀q ∈ Qh.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The mimetic conservation equation: DIVhFh = bI

Take the cell average of the conservation equation: div − → F = b − → 1 |E|

E

div − → F dV = 1 |E|

E

b dV = bI|E and define the flux Fh as the solution to

DIVhFh)E = bI|E ≡ 1

|E|

E

div − → F dV using the inner product

p, q
Qh :=

E |E| pEqE =

Ω

p q dV, variational form:

DIVhFh, q
Qh =
bI, q
Qh

∀q ∈ Qh.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The mimetic constitutive equation: Fh = Ghph

Now, we discretize the constitutive equation: − → F = −K∇p Note that the conservation equation 1 |E|

e∈∂E

|e| (Fh)e

E = (bI)E

∀E ∈ Th holds for

many finite volume schemes;
the RT0−P0 mixed finite element method

. . . So let us first have a look at these approaches.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The mimetic constitutive equation: Fh = Ghph

Now, we discretize the constitutive equation: − → F = −K∇p Note that the conservation equation 1 |E|

e∈∂E

|e| (Fh)e

E = (bI)E

∀E ∈ Th holds for

many finite volume schemes;
the RT0−P0 mixed finite element method

. . . So let us first have a look at these approaches.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Finite volume discretization

We use a direct formula for the flux: − → F = −K∇p (with K = κI)

K

L A B − → n AB −− → n AB·− → F ≈ κ pK − pL

−

→ x K − − → x L

K

L A B −− → F ≈ κ

pK − pL
−

→ x K − − → x L

−

→ n AB γAB + pB − pA

−

→ x B − − → x A

−

→ n KL γKL

Manzini, G.

The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

In mixed finite element the constitutive equation is discretized by using an explicit representation of the flux field from the flux degrees of freedom inside each element. Let E be a triangle and take the Raviart-Thomas space: RT0(E) :=

−

→ v (x, y) = α β

+ γ

x y

(x, y) ∈ E
.

Reconstruct RE (G) inside E by using the degrees of freedom Ge

E and the canonical basis functions of RT0(E).

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

In mixed finite element the constitutive equation is discretized by using an explicit representation of the flux field from the flux degrees of freedom inside each element. Let E be a triangle and take the Raviart-Thomas space: RT0(E) :=

−

→ v (x, y) = α β

+ γ

x y

(x, y) ∈ E
.

Reconstruct RE (G) inside E by using the degrees of freedom Ge

E and the canonical basis functions of RT0(E).

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

The Raviart-Thomas field RE (G) preserves:

1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E

2. the elemental divergence: div RE
G
= DIVh,EG
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c .

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

The Raviart-Thomas field RE (G) preserves:

1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E

2. the elemental divergence: div RE
G
= DIVh,EG
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c .

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

The Raviart-Thomas field RE (G) preserves:

1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E

2. the elemental divergence: div RE
G
= DIVh,EG
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c .

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

Given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the “RT0 inner product ”:
Fh, G
RT0 :=
E
E

K−1RE (Fh) · RE (G) dV

3. rewrite the RT0 − P0 variational form of

K−1− → F = −∇p as:

Fh, G
RT0 =
E
E

ph div RE (G) dV ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

Given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the “RT0 inner product ”:
Fh, G
RT0 :=
E
E

K−1RE (Fh) · RE (G) dV

3. rewrite the RT0 − P0 variational form of

K−1− → F = −∇p as:

Fh, G
RT0 =
E
E

ph div RE (G) dV ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mixed finite elements discretization

Given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the “RT0 inner product ”:
Fh, G
RT0 :=
E
E

K−1RE (Fh) · RE (G) dV

3. rewrite the RT0 − P0 variational form of

K−1− → F = −∇p as:

Fh, G
RT0 =
E
E

ph div RE (G) dV ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

FIRST, let RE

·
be a reconstruction for E with the properties
f mixed finite elements:
1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E;

2. the elemental divergence: div RE
G
= DIVh,EG;
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c . We can build many operators: we do not have uniqueness!

Manzini, G. The Mimetic Finite Difference Method

SLIDE 38

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

FIRST, let RE

·
be a reconstruction for E with the properties
f mixed finite elements:
1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E;

2. the elemental divergence: div RE
G
= DIVh,EG;
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c . We can build many operators: we do not have uniqueness!

Manzini, G. The Mimetic Finite Difference Method

SLIDE 39

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

FIRST, let RE

·
be a reconstruction for E with the properties
f mixed finite elements:
1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E;

2. the elemental divergence: div RE
G
= DIVh,EG;
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c . We can build many operators: we do not have uniqueness!

Manzini, G. The Mimetic Finite Difference Method

SLIDE 40

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

FIRST, let RE

·
be a reconstruction for E with the properties
f mixed finite elements:
1. the degrees of freedom:

− → n e

E · RE

− → G

= Ge

E;

2. the elemental divergence: div RE
G
= DIVh,EG;
3. constant vector fields: P0-compatible;

let G

− → c = (−

→ c )I with − → c constant inside E; then, RE

G

− → c

= − → c . We can build many operators: we do not have uniqueness!

Manzini, G. The Mimetic Finite Difference Method

SLIDE 41

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

THEN, given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the mimetic inner product:
Fh, G
Xh :=
E
E

K−1RE (Fh) · RE (G) dV

3. mimetic discretization of

K−1− → F = −∇p:

Fh, G
Xh =
E
E

phdiv RE (G)dV =

ph, DIVhG
Qh ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

SLIDE 42

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

THEN, given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the mimetic inner product:
Fh, G
Xh :=
E
E

K−1RE (Fh) · RE (G) dV

3. mimetic discretization of

K−1− → F = −∇p:

Fh, G
Xh =
E
E

phdiv RE (G)dV =

ph, DIVhG
Qh ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

SLIDE 43

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

MFD: mimicking Mixed Finite Elements

THEN, given Fh, G in Xh (degrees of freedom):

1. reconstruct RE (Fh), RE (G) inside each element E;
2. define the mimetic inner product:
Fh, G
Xh :=
E
E

K−1RE (Fh) · RE (G) dV

3. mimetic discretization of

K−1− → F = −∇p:

Fh, G
Xh =
E
E

phdiv RE (G)dV =

ph, DIVhG
Qh ∀G ∈ Xh

Manzini, G. The Mimetic Finite Difference Method

SLIDE 44

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The Mimetic Finite Difference Method

Scheme formulation:

Fh, G
Xh −
ph, DIVhG
Qh

= ∀G ∈ Xh

DIVhFh, q
Qh

=

bI, q
Qh

∀q ∈ Qh.

Substitute Fh = Ghph in the first equation:
Ghph, G
Xh =
ph, DIVhG
Qh

∀G ∈ Xh to get the MFD discretization: K−1− → F = −∇p − → Fh = Ghph in Xh

Manzini, G. The Mimetic Finite Difference Method

SLIDE 45

Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

The Mimetic Finite Difference Method

Scheme formulation:

Fh, G
Xh −
ph, DIVhG
Qh

= ∀G ∈ Xh

DIVhFh, q
Qh

=

bI, q
Qh

∀q ∈ Qh.

Substitute Fh = Ghph in the first equation:
Ghph, G
Xh =
ph, DIVhG
Qh

∀G ∈ Xh to get the MFD discretization: K−1− → F = −∇p − → Fh = Ghph in Xh

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Features

Let R(·) = {RE (·)} be a lifting operator such that RE (·) is P0-compatible on E.

1. R(Xh) ⊂ H(div, Ω); hence R(Xh) − P0 is a conforming

mixed discretization that generalizes RT0 − P0;

2. we can use elements of very general shapes, even

non-convex (but at least star-shaped) elements are admissible;

3. the implementation and analysis of the MFD method are

similar to those of the Mixed Finite Element method.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary Formal construction Mimetic conservation equation Mimetic constitutive equation

Mimetic scalar product for the flux

The scalar product for fluxes is given by assembling “elemental” scalar products, each one of which can be represented by a symmetric positive definite (SPD) matrix:

E

K−1RE (Fh) · RE (G) dV − → ME = RE K−1

E

|E| RT

E +

ME RE

·
is not unique

− → family of matrices

ME contains free positive parameters and ensures that ME

is an SPD matrix; the formulas for RE, ME depend on the shape of E and can be derived without the explicit knowledge of RE

.

Manzini, G. The Mimetic Finite Difference Method

SLIDE 48

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

A priori error estimates for Darcy’s equation

[Brezzi-Lipnikov-Shashkov, SINUM 2005]

General assumptions:

Ω polygonal or polyhedral with Lipschitz continuous boundary;
Th is a partition satisfying some mesh regularity assumptions;
the scalar product
·, ·
Xh satisfies local consistency and stability;
1. If p ∈ H2(Ω) then
−

→ F I − Fh

Xh ≤ Ch
p
H2(Ω)

where

·
2

Xh =

·, ·
Xh.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

A priori error estimates for Darcy’s equation

[Brezzi-Lipnikov-Shashkov, SINUM 2005]

2. if p ∈ H2(Ω) then
pI − ph
Qh ≤ Ch
p
H2(Ω)
3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex:
pI − ph
Qh ≤ Ch2
p
H2(Ω) +
b
H1(Ω)
Manzini, G.

The Mimetic Finite Difference Method

SLIDE 51

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

A priori error estimates for Darcy’s equation

[Brezzi-Lipnikov-Shashkov, SINUM 2005]

2. if p ∈ H2(Ω) then
pI − ph
Qh ≤ Ch
p
H2(Ω)
3. superconvergence: if p∈H2(Ω), b∈H1(Ω), and Ω is convex:
pI − ph
Qh ≤ Ch2
p
H2(Ω) +
b
H1(Ω)
Manzini, G.

The Mimetic Finite Difference Method

SLIDE 52

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Post-processing of solution

[Cangiani-M. CMAME 2008]

Let − → v be a vector field on E. Then, − → v

Interpolate

− → − → v I

Reconstruct

− → − → v ∗ ∈

P0(E)

d where the elemental reconstructed vector is given by:

E

− → v ∗ · ∇q = − → v I, (KE∇q)I

E

∀q ∈ P1(E). In effect, − → v ∗ = RT

E−

→ v I where RE is the matrix used in the definition of the elemental scalar product; so, easy to implement and very cheap.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Post-processing of solution

[Cangiani-M. CMAME 2008, L. Beirao da Veiga Numer. Math. 2008]

1. Reconstruct the elemental gradient:

Fh|E

Reconstruct

− → − → F ∗

h|E Calculate

− → ∇Ep∗

h = −K−1 E

− → F ∗

h;

2. reconstruct the piecewise linear pressure field as

ph|∗

E(−

→ x ) := pE + ∇Ep∗

h · (−

→ x − − → x E), ∀− → x ∈ E (− → x E center of gravity of E) Under the hypothesis yielding scalar super-convergence:

p − p∗

h

L2(Ω) + h
p − p∗

h

1,h ≤ Ch2
p
H2(Ω) +
b
H1(Ω)
with
q
2

1,h =

E
∇q
2

L2(E) +

e

h−1

e

[

[q] ]

2

e.

Manzini, G. The Mimetic Finite Difference Method

SLIDE 55

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

SLIDE 56

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

An a posteriori error estimator for MFDs

[L. Beirao da Veiga, Numer. Math. 2008]

p∗

h, post-processed pressure; [

[q] ]e jump of q across edge e. error estimator: η2 :=

E ηE 2 and

ηE

2 :=

Fh + (KE∇p∗

h)I

2

E + 1

2

e∈∂E

h−1

e

[

[p∗

h]

]e

2

L2(e)

target error: err2 =

E errE 2 with

errE

2 :=

−

→ F − R(Fh)

2

L2(E) + h2 E

div (−

→ F − R(Fh))

2

L2(E) +

p − p∗

h

1,E

and

q
2

1,E =

∇q
2

L2(E) +

e∈∂E

h−1

e

[

[q] ]e

2

L2(e)

efficiency: cηE ≤ errE; reliability: err ≤ Cη;.

Manzini, G. The Mimetic Finite Difference Method

SLIDE 57

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

An a posteriori error estimator for MFDs

[L. Beirao da Veiga, Numer. Math. 2008]

p∗

h, post-processed pressure; [

[q] ]e jump of q across edge e. error estimator: η2 :=

E ηE 2 and

ηE

2 :=

Fh + (KE∇p∗

h)I

2

E + 1

2

e∈∂E

h−1

e

[

[p∗

h]

]e

2

L2(e)

target error: err2 =

E errE 2 with

errE

2 :=

−

→ F − R(Fh)

2

L2(E) + h2 E

div (−

→ F − R(Fh))

2

L2(E) +

p − p∗

h

1,E

and

q
2

1,E =

∇q
2

L2(E) +

e∈∂E

h−1

e

[

[q] ]e

2

L2(e)

efficiency: cηE ≤ errE; reliability: err ≤ Cη;.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Some experiments on mesh adaptivity

[L. Beirao da Veiga & M., IJNME, 2008]

The indicator ηE allows us to select elements needing refinement:

1. calculate local error estimates ηE;
2. mark for refinement those elements such that ηE ≥ tol ηmax;
3. subdivide the marked elements as follows:
No need for special treatment of hanging nodes!

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Mesh adaptivity

L-shaped domain, K = I I, p(r, θ) = r 2/3 sin(2θ/3)

10

2

10

3

10

4

10

5

10

6

10

3

10

2

10

1

10

After 3 refinements η(◦), err(•) vs #elements

Manzini, G. The Mimetic Finite Difference Method

SLIDE 60

Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Outline

1

MFD method for Darcy’s problem Formal construction Mimetic conservation equation Mimetic constitutive equation

2

Theoretical results and applications A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

3

Summary

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Convection-Diffusion-Reaction Equation

(In collaboration with A. Cangiani and A. Russo)

Problem:

− → F = −(K ∇ p + − → β p) in Ω div − → F + c p = b in Ω p = 0

n ∂Ω

with

β, K and c smooth fields plus usual coercivity conditions,
Ω ⊂ I

R2 is a polygonal domain.

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Convection-Diffusion-Reaction Equation

Scheme formulation

discretization of convection-reaction terms convection:

Ω

K−1− → β p · R

G
dV

− →

E pE

(−

→ β )I, G

E

reaction:

Ω

c p q dV − →

cIph, q
Qh

Scheme formulation

Fh, G
Xh −
p, DIVhG
Qh +

E pE

(−

→ β )I, G

E = 0

∀G ∈ Xh

DIVhFh, q
Qh +
cIph, q
Qh =
b, q
Qh

∀q ∈ Qh

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Convection-Diffusion-Reaction Equation

a priori estimate for the diffusive regime:

−

→ F − R(Fh)

H(div,Ω) +
p − ph
L2(Ω)

≤ Ch

p
H1(Ω) + h
p
H2(Ω) +
b − bI
L2(Ω)
,

(with − → β ∈ W 2,∞(Ω), c ∈ W 1,∞(Ω), and coercivity assumptions) superconvergence for pressure cell averages under same assumptions of the pure elliptic problem and − → β ∈ R(Xh):

pI − ph
Qh ≤ Ch2
p
H2(Ω) +
b
H1(Ω)
Manzini, G.

The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Convection-Diffusion-Reaction Equation

Convergence results p(x, y) = sin(2π x) sin(2π y) +x2 + y2 + 1, − → β = (1, 3)T, c(x, y) = x y2 n h L2-error Rate Hdiv-error Rate 1 9.135 10−2 9.134 10−2 −− 1.823 10−1 −− 2 4.654 10−2 4.630 10−2 1.007 8.572 10−2 1.118 3 2.346 10−2 2.315 10−2 1.012 4.168 10−2 1.052 4 1.175 10−2 1.164 10−2 0.995 2.039 10−2 1.034 5 5.880 10−3 5.841 10−3 0.995 1.007 10−2 1.018 6 2.940 10−3 2.927 10−3 0.996 5.027 10−3 1.002

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation

Convection-Diffusion-Reaction Equation

Superconvergence results p(x, y) = sin(2π x) sin(2π y) +x2 + y2 + 1, − → β = (1, 3)T, c(x, y) = x y2 n h Qh-error Rate 1 9.135 10−2 3.069 10−2 −− 2 4.654 10−2 1.078 10−2 1.551 3 2.346 10−2 2.807 10−3 1.964 4 1.175 10−2 7.483 10−4 1.913 5 5.880 10−3 1.904 10−4 1.975 6 2.940 10−3 4.796 10−5 1.989

Manzini, G. The Mimetic Finite Difference Method

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Mimetic Formulation Theoretical results and applications Summary

Summary

The MFD method for second-order elliptic problems mimics properties of continuous operators; e.g. DIVh, Gh satisfy discrete Green-like formulas; works on element of very general shape; shows a strong connection with the lowest-order mixed finite element method RT0 − P0, helpful in establishing the theoretical foundation.

Manzini, G. The Mimetic Finite Difference Method