[PPT] - Mimetic scalar products of discrete differential forms Annalisa PowerPoint Presentation

SLIDE 1

Mimetic scalar products of discrete differential forms

Annalisa Buffa

IMATI “E. Magenes” - Pavia Consiglio Nazionale delle Ricerche

In collaboration with F. Brezzi and M. Manzini

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 1 / 25

SLIDE 2

1

What is this talk about

2

Differential forms and Finite Elements

3

Differential forms and Mimetic Finite Differences

4

Open problems - Conclusions

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 2 / 25

SLIDE 3

The aim of the talk

We are interested in providing numerical solutions to PDEs whose unknown can be sought as differential forms Examples are diffusion, fluid flows, elasticity, electromagnetics... There are many attempts in the literature !

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 3 / 25

SLIDE 4

The aim of the talk

We are interested in providing numerical solutions to PDEs whose unknown can be sought as differential forms Examples are diffusion, fluid flows, elasticity, electromagnetics... There are many attempts in the literature !

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 3 / 25

SLIDE 5

The aim of the talk

We are interested in providing numerical solutions to PDEs whose unknown can be sought as differential forms Examples are diffusion, fluid flows, elasticity, electromagnetics... There are many attempts in the literature !

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 3 / 25

SLIDE 6

The aim of the talk

We are interested in providing numerical solutions to PDEs whose unknown can be sought as differential forms Examples are diffusion, fluid flows, elasticity, electromagnetics... There are many attempts in the literature ! We consider here and all along the talk equations of electrostatics/magnetostatics as “toy” problems: Ω ⊂ Rd, bounded, Lipschitz.    div a u = g Ω curl u = f Ω a u · n = 0 ∂Ω.

Ω n

In what follows: Ω is simply connected.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 3 / 25

SLIDE 7

Our example

Ω ⊂ R3, bounded, Lipschitz.    div a u = g Ω curl u = f Ω + B.C.

Ω n

u ∈ Λ1, ⋆a : Λ1 → Λ2    d ⋆a u = g d u = f Tr(⋆au) = 0. If f = 0, u = d χ, χ ∈ Λ0 and d ⋆a d χ = g v ∈ Λ2, ⋆a−1 : Λ2 → Λ1    d v = g d ⋆a−1 v = f Tr(v) = 0. If g = 0, v = d ξ, ξ ∈ Λ1 and d ⋆a−1 d ξ = f

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 4 / 25

SLIDE 8

De-Rham diagram

Solvability is due to the (trivial) De-Rham complex: when Ω is simply connected 0 → Λ0

d

− − − − → Λ1

d

− − − − → Λ2

d

− − − − → Λ3 → 0 On proxy fields this is 0 → H1 (Ω)

∇

− − − − → H (curl , Ω)

curl

− − − − → H (div, Ω)

div

− − − − → L2 (Ω) → 0 with H (curl , Ω) = {u ∈ L2(Ω)2 : curl u ∈ L2(Ω)3 } H (div, Ω) = {u ∈ L2(Ω)2 : divu ∈ L2(Ω) }

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 5 / 25

SLIDE 9

De-Rham diagram

Solvability is due to the (trivial) De-Rham complex: when Ω is simply connected 0 → Λ0

d

− − − − → Λ1

d

− − − − → Λ2

d

− − − − → Λ3 → 0 On proxy fields this is 0 → H10(Ω)

∇

− − − − → H0(curl , Ω)

curl

− − − − → H0(div, Ω)

div

− − − − → L20(Ω) → 0 with H0(curl , Ω) = {u ∈ L2(Ω)2 : curl u ∈ L2(Ω)3 u × n = 0 on ∂Ω} H0(div, Ω) = {u ∈ L2(Ω)2 : divu ∈ L2(Ω) u · n = 0 on ∂Ω} L2

0(Ω) = {u ∈ L2(Ω) :

Ω u = 0}
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 5 / 25

SLIDE 10

Variational formulations: proxy fields

All those problems admits a variational formulations:

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 6 / 25

SLIDE 11

Variational formulations: proxy fields

All those problems admits a variational formulations: if f = 0, we just wrote the equation for the electric scalar potential χ ∈ Λ0 : d ⋆a dχ = g. On the proxy field: χ ∈ H1(Ω) :

Ω

a ∇χ · ∇χt =

Ω

g χt ∀ χt ∈ H1(Ω). Alternatively, setting D = a∇χ: Find D ∈ H0(div, Ω), χ ∈ L2

0(Ω)

Ω

a−1D · Dt +

Ω

χ divDt = 0 ∀Dt

Ω

divD χt =

Ω

gχt ∀χt.

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 6 / 25

SLIDE 12

Variational formulations: proxy fields

All those problems admits a variational formulations: if g = 0, we just wrote v = dξ, ξ ∈ Λ1 is the magnetic vector potential and verifies d ⋆a−1 dξ = f which on the proxy field is: find ξ ∈ H(curl , Ω), p ∈ H1(Ω)

Ω

a−1 curl ξ · curl ξt −

Ω

∇p · ξt =

Ω

f · ξt ∀ ξt

Ω

ξ · ∇q = 0 ∀q.

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 6 / 25

SLIDE 13

Compatible discretization

Mimic the geometric structure at the discrete level

The idea is to construct discretizations which mimic, or preserve the geometric structure...

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 7 / 25

SLIDE 14

Compatible discretization

Mimic the geometric structure at the discrete level

The idea is to construct discretizations which mimic, or preserve the geometric structure... Finite Elements Techniques

Arnold et al, 2006 , Bossavit ’87-, Hiptmair ’99-, Dular 90s-, Demkowicz ’98- . . . , Christiansen 05-, B. 05-

Mimetic Finite Differences

Hyman-Shashkov ’90s, Brezzi et al. ’04-’05, B.’07

Finite Volumes, Finite Differences, Primal-Dual methods, Finite Integration Techniques

Yee ’66, Bossavit-Kettunen ’99-, Tonti 95- Weiland ’77-, Rain ’02-’07, Bochev ’04- . . . ,

And for a pure “geometric” approach: Electromagnetic theory on lattices Teixeira-Chew, ’99 Discrete exterior calculus Leok et al. ’04- Applied differential geometry Bossavit, Kotiuga

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 7 / 25

SLIDE 15

From continuous to discrete

The De Rham map

Polyhedral subdivision

f the domain:
(C0, C1, C2, C3), chain complex, denote

the set of vertices , edges , faces and elements , all endowed with an orientation.

(C 0, C 1, C 2, C 3) cochain complex:

functionals on Ci, i = 0, . . . 3

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 8 / 25

SLIDE 16

From continuous to discrete

The De Rham map

Polyhedral subdivision

f the domain:
(C0, C1, C2, C3), chain complex, denote

the set of vertices , edges , faces and elements , all endowed with an orientation.

(C 0, C 1, C 2, C 3) cochain complex:

functionals on Ci, i = 0, . . . 3 There is a standard way to go from Λi to C i: the De Rham map Πi, i = 0, . . . 3 ui ∈ Λi Πi(ui) ∈ C i Πi(ui) =

ci ui , ci ∈ Ci
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 8 / 25

SLIDE 17

From continuous to discrete

The De Rham map

Polyhedral subdivision

f the domain:
(C0, C1, C2, C3), chain complex, denote

the set of vertices , edges , faces and elements , all endowed with an orientation.

(C 0, C 1, C 2, C 3) cochain complex:

functionals on Ci, i = 0, . . . 3 There is a standard way to go from Λi to C i: the De Rham map Πi, i = 0, . . . 3 u1 ∈ Λ1 Π1(u1) ∈ C 1 Π1(u1) =

e u1 , e ∈ C1
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 8 / 25

SLIDE 18

From continuous to discrete

The De Rham map

Polyhedral subdivision

f the domain:
(C0, C1, C2, C3), chain complex, denote

the set of vertices , edges , faces and elements , all endowed with an orientation.

(C 0, C 1, C 2, C 3) cochain complex:

functionals on Ci, i = 0, . . . 3 There is a standard way to go from Λi to C i: the De Rham map Πi, i = 0, . . . 3 u1 ∈ Λ1 Π1(u1) ∈ C 1 Π1(u1) =

e u1 , e ∈ C1
Approximation of Differential forms

C i are chosen as approximations of differential forms Λi.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 8 / 25

SLIDE 19

How to use cochains as discretizations

C 0 nodal values of the unknown Λ0 at vertices C 1 circulations of the unknown Λ1 along edges C 2 fluxes of the unknown Λ2 through faces C 3 average of the unknown Λ3 on elements

Main requirements: Differential operators : we need to compute all first order differential operators; Scalar products : we need L2-like scalar products to implement

ur variational principles
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 9 / 25

SLIDE 20

Coboundary operators as differential operators

Once the orientation of edges , faces and elements is fixed, we have: GRAD :C 0 → C 1

GRAD u

  

e = u|V 1 − u|V 2

1 2

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 10 / 25

SLIDE 21

Coboundary operators as differential operators

Once the orientation of edges , faces and elements is fixed, we have: GRAD :C 0 → C 1

GRAD u

  

e = u|V 1 − u|V 2

CURL :C 1 → C 2

CURLφ

  

e∈f =

i=1

δ(e, f ) φ|e

1 2

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 10 / 25

SLIDE 22

Coboundary operators as differential operators

Once the orientation of edges , faces and elements is fixed, we have: GRAD :C 0 → C 1

GRAD u

  

e = u|V 1 − u|V 2

CURL :C 1 → C 2

CURLφ

  

e∈f =

i=1

δ(e, f ) φ|e DIV :C 2 → C 3

DIVσ

  

P =

f ∈P

δ(f , P) σ|f

1 2

back

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 10 / 25

SLIDE 23

Construction of a discretization: simplicial meshes

Finite elements

On simplicial meshes:

C 0

a + c · x P1 elements

C 1

a + c ∧ x Edge elements

C 2

a + c x Face Elements

C 3

P0 elements

From cochains to finite elements

Finite elements are piecewise polynomial reconstructions of cochains and we use finite elements instead of cochains! Reconstructions : Ri : C i → X i. It is onto and RiΠi = Idi on X i

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 11 / 25

SLIDE 24

Construction of a discretization: simplicial meshes

Finite elements

On simplicial meshes:

C 0

a + c · x P1 elements

C 1

a + c ∧ x Edge elements

C 2

a + c x Face Elements

C 3

P0 elements

From cochains to finite elements

Finite elements are piecewise polynomial reconstructions of cochains and we use finite elements instead of cochains! Reconstructions : Ri : C i → X i. It is onto and RiΠi = Idi on X i Scalar product : standard scalar products between functions in L2!

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 11 / 25

SLIDE 25

Construction of a discretization: simplicial meshes

We can apply the variational principle and obtain discretizations. Known Results: The obtained finite element methods are optimally convergent, spectrally correct and so on..

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 12 / 25

SLIDE 26

Construction of a discretization: general meshes

Mimetic Finite Differences, revisited

for each C i, we need to choose scalar products which have to mimic the L2 scalar products on proxy fields.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 13 / 25

SLIDE 27

Construction of a discretization: general meshes

Mimetic Finite Differences, revisited

for each C i, we need to choose scalar products which have to mimic the L2 scalar products on proxy fields. First of all: this can be done element by element [u, v]i =

P∈Th

[u|P, v|P]P,i ∀ u , v ∈ C i We have at least two possibilities:

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 13 / 25

SLIDE 28

Construction of a discretization: general meshes

Mimetic Finite Differences, revisited

for each C i, we need to choose scalar products which have to mimic the L2 scalar products on proxy fields. First of all: this can be done element by element [u, v]i =

P∈Th

[u|P, v|P]P,i ∀ u , v ∈ C i We have at least two possibilities: For each P, define reconstructions Ri and set: [u, v]P,i =

P

Ri(u) · Ri(v)dx ∀ u , v ∈ C i|P. Can/must be done with submeshing: demanding!

Kuznetsov et al ’03 Christiansen ’07

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 13 / 25

SLIDE 29

Construction of a discretization: general meshes

Mimetic Finite Differences, revisited

for each C i, we need to choose scalar products which have to mimic the L2 scalar products on proxy fields. First of all: this can be done element by element [u, v]i =

P∈Th

[u|P, v|P]P,i ∀ u , v ∈ C i We have at least two possibilities: For each P, define reconstructions Ri and set: [u, v]P,i =

P

Ri(u) · Ri(v)dx ∀ u , v ∈ C i|P. Can/must be done with submeshing: demanding!

Kuznetsov et al ’03 Christiansen ’07

What can we say without having reconstructions at hand? How exact must be the scalar products to ensure optimal convergence? (Mimetic FD approach)

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 13 / 25

SLIDE 30

Scalar products on a single element: requirements

Given an element P with N vertices, E edges and F faces, we need three ”mass” matrices: M0 ∈ RN×N M1 ∈ RE×E M2 ∈ RF×F The M3 is a number (one unknown per element) which can be safely chosen to be equal to |P| !

The requirement : The scalar products must be P0-exact, i.e., must

be exact when applied to a generic cochain and a cochain which is the interpolation (De Rham map)

f a constant field.
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 14 / 25

SLIDE 31

Scalar products on a single element: requirements

Given an element P with N vertices, E edges and F faces, we need three ”mass” matrices: M0 ∈ RN×N M1 ∈ RE×E M2 ∈ RF×F The M3 is a number (one unknown per element) which can be safely chosen to be equal to |P| !

The requirement : The scalar products must be P0-exact, i.e., must

be exact when applied to a generic cochain and a cochain which is the interpolation (De Rham map)

f a constant field.

The Idea : If we get constants right, we have a first order scheme

...

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 14 / 25

SLIDE 32

Meaning of being P0-exact

Let us focus on [·, ·]P,1, i.e., scalar products on C 1, i.e., scalar products among “circulations” over edges. We work now on a single element P ∈ Th.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 15 / 25

SLIDE 33

Meaning of being P0-exact

Let us focus on [·, ·]P,1, i.e., scalar products on C 1, i.e., scalar products among “circulations” over edges. We work now on a single element P ∈ Th. A reconstruction R1 : C 1(P) → H(curl , P) is P0-preserving, if:

✞ ✝ ☎ ✆

R1(Π1c) = c , if CURL u = Π2c′ then curl R1(u) = c′

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 15 / 25

SLIDE 34

Meaning of being P0-exact

Let us focus on [·, ·]P,1, i.e., scalar products on C 1, i.e., scalar products among “circulations” over edges. We work now on a single element P ∈ Th. A reconstruction R1 : C 1(P) → H(curl , P) is P0-preserving, if:

✞ ✝ ☎ ✆

R1(Π1c) = c , if CURL u = Π2c′ then curl R1(u) = c′

P0-exact

[·, ·]P,1 is P0-exact if there exists at least one P0-preserving reconstruction R1 such that for a generic u ∈ C 1 : [u, Π1c]P,1 :=

P R1(u) · c
A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 15 / 25

SLIDE 35

Examples of “good“ reconstructions

Technical slide ...

Let us start from u ∈ C 1: we have circulations over edges of R1(u). We reconstruct first on face, and then on elements. On each face: 1. curlf R1f (u) = 1 |f |(CURLu)|f

2. divf R1f (u) = 0
3. R1f (u) · t = 1

|e|u|e ∀e ∈ ∂f On the element P: 1. curl curl R1(u) = 0

2. divR1(u) = 0
3. R1(u)T,f = R1f (u)

∀f ∈ ∂P Then, one can integrate by parts a few times..

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 16 / 25

SLIDE 36

Examples of “good“ reconstructions -II

Then, one can integrate by parts a few times.. x running coordinates on the element, xP barycenter of P

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 17 / 25

SLIDE 37

Examples of “good“ reconstructions -II

Then, one can integrate by parts a few times.. x running coordinates on the element, xP barycenter of P ξ running coordinates on the face, ξf barycenter of the face f

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 17 / 25

SLIDE 38

Examples of “good“ reconstructions -II

Then, one can integrate by parts a few times.. x running coordinates on the element, xP barycenter of P ξ running coordinates on the face, ξf barycenter of the face f (a1, a2)⊥ = (−a2, a1) [Π1c, u]P,1 :=

P

R1(u) · c = −

f ∈∂P
e∈∂f

1 |e|u|e

e

α⊥

f · (ξ − ξf )

where we have set: αf = (nf · (x − xP)) cT + (nf · c) ((xP)T,f − ξf )

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 17 / 25

SLIDE 39

Examples of “good“ reconstructions -II

Then, one can integrate by parts a few times.. x running coordinates on the element, xP barycenter of P ξ running coordinates on the face, ξf barycenter of the face f (a1, a2)⊥ = (−a2, a1) [Π1c, u]P,1 :=

P

R1(u) · c = −

f ∈∂P
e∈∂f

1 |e|u|e

e

α⊥

f · (ξ − ξf )

where we have set: αf = (nf · (x − xP)) cT + (nf · c) ((xP)T,f − ξf ) This is true on any polyhedron! Computationally, we need to know the barycenters of elements and faces, and to be able to compute integrals along edges of P1 polynomials!

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 17 / 25

SLIDE 40

Class of Reconstructions

We can repeat the reasoning for all C i There are entire classes of reconstructions with the same average... Indeed, the natural way to construct those classes is to ask P0-preserving properties and commuting properties as: R0(C 0)

∇

− − − − → R1(C 1)

curl

− − − − → R2(C 2)

div

− − − − → R3(C 3)

R0





R1





R2





R3



 C 0

GRAD

− − − − → C 1

CURL

− − − − → C 2

DIV

− − − − → C 3 In other words: e.g., ∇R0(u) = R1(GRADu) curl R1(φ) = R2(CURLφ) divR2(σ) = R3(DIVσ)

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 18 / 25

SLIDE 41

Scalar products

We have a formula for [Π1c, u]P,1. This gives information only on a part of the scalar product...

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 19 / 25

SLIDE 42

Scalar products

We have a formula for [Π1c, u]P,1. This gives information only on a part of the scalar product...

Question : how do we complete the scalar products? Answer : with any matrix having the right scaling!

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 19 / 25

SLIDE 43

Scalar products

Consider again [·, ·]P,1. Basis: φ1 = Π1(e1) , φ2 = Π1(e2) , φ3 = Π1(e3), φ4, φ5 . . . φE

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 20 / 25

SLIDE 44

Scalar products

Consider again [·, ·]P,1. Basis: φ1 = Π1(e1) , φ2 = Π1(e2) , φ3 = Π1(e3), φ4, φ5 . . . φE For example, M1 ⇒

3 E

1 2 3 4 . . .

3 3 x

α I φ φ φ φ

Ε

φ where α ∼ diam(P) (scaling factor of the pull-back operator).

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 20 / 25

SLIDE 45

Scalar products

Consider again [·, ·]P,1. Basis: φ1 = Π1(e1) , φ2 = Π1(e2) , φ3 = Π1(e3), φ4, φ5 . . . φE For example, M1 ⇒

3 E

1 2 3 4 . . .

3 3 x

α I φ φ φ φ

Ε

φ where α ∼ diam(P) (scaling factor of the pull-back operator). I ⇔ any symmetric positive definite matrix with bounded eigenvalues!

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 20 / 25

SLIDE 46

A useful property

Given a scalar product on an element P , [u, v]P,1, i.e. we have chosen a suitably scaled matrix THEN there exists a P0-preserving R1 reconstruction such that: [u, v]P,1 =

P

R1(u) · R1(v) ∀ u , v ∈ C 1.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 21 / 25

SLIDE 47

A useful property

Given a scalar product on an element P , [u, v]P,1, i.e. we have chosen a suitably scaled matrix THEN there exists a P0-preserving R1 reconstruction such that: [u, v]P,1 =

P

R1(u) · R1(v) ∀ u , v ∈ C 1. This is uninteresting for numerics, but useful to develop convergence analysis...

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 21 / 25

SLIDE 48

Discrete problems

We are now ready to formulate our discrete variational principles:

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 22 / 25

SLIDE 49

Discrete problems

We are now ready to formulate our discrete variational principles: The equation: χ ∈ H1(Ω) :

Ω a ∇χ · ∇χt =
Ω g χt.

becomes: with · denoting the projection onto piecewise constants χh ∈ C 0 : [ a GRADχh, GRADχt

h]1 =

P

[ g, χt

h]0

∀χt

h

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 22 / 25

SLIDE 50

Discrete problems

We are now ready to formulate our discrete variational principles: The problem: Find D ∈ H0(div, Ω), χ ∈ L2

0(Ω)

Ω

a−1D · Dt +

Ω

χ divD = 0 ∀Dt

Ω

divD χt =

Ω

gχt ∀χt. becomes: Find Dh ∈ C 2

0 , χh ∈ C 3 0 :

[ a−1Dh, Dt

h]2 + [χh, DIV Dt h]3 = 0

∀Dt

h

[DIV Dh, χt

h]3 = [

g, χt

h]3

∀χt

h.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 22 / 25

SLIDE 51

Discrete problems

We are now ready to formulate our discrete variational principles: The problem: find ξ ∈ H(curl , Ω), p ∈ H1(Ω)

Ω

a−1curl ξ · curl ξt −

Ω

∇p · ξt =

Ω

f · ξt ∀ ξt

Ω

ξ · ∇q = 0 ∀q. becomes: find ξh ∈ C 1, ph ∈ C 0 : [ a−1CURL ξh, CURL ξt

h]2 − [GRAD ph, ξt h]1 =

P

[ f, ξt]P,1 ∀ ξt

h

[ξh, GRAD qh]1 = 0 ∀qh.

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 22 / 25

SLIDE 52

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 23 / 25

SLIDE 53

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

Exact Computed Slope n =4 8 16 32 D.o.f. 40 144 544 2112 Mainly-hexagonal mesh 2 2,498 11 2,127 98 2,029 16 2,006 94 2.21 5 7,480 37 5,654 46 5,156 30 5,037 60 2.20 5 11,222 30 6,054 87 5,212 12 5,048 52 2.28 8 17,399 60 10,387 30 8,518 53 8,114 92 2.33 10 35,366 70 13,754 70 10,750 10 10,171 50 2.28 10 39,303 00 14,018 10 10,757 70 10,173 20 2.28 13 44,691 00 18,230 60 14,151 80 13,259 40 2.30 13 91,493 50 23,781 00 14,807 80 13,361 60 2.48 D.o.f. 88 272 928 3392

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 23 / 25

SLIDE 54

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

Exact Computed Slope n =4 8 16 32 D.o.f. 88 272 928 3392 Non-convex mesh 2 5,449 54 2,838 70 2,207 36 2,051 55 2.05 5 21,809 70 8,660 32 5,889 14 5,220 33 2.06 5 22,053 30 8,687 31 5,890 81 5,220 34 2.06 8 68,435 80 20,001 50 11,345 10 8,826 47 2.06 10 79,732 20 20,167 70 12,300 20 10,564 60 2.07 10 81,477 20 21,866 30 12,311 80 10,565 30 2.08 13 169,341 00 41,244 80 20,743 30 14,901 80 2.07 13 172,088 00 41,355 70 20,751 90 14,902 00 2.07 D.o.f. 80 288 1088 4224

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 23 / 25

SLIDE 55

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 24 / 25

SLIDE 56

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

Exact Computed Slope

ref. =1

2 3 4

ref. =1

2 3 4 Cubic mesh 9.639724 12,881 20 10,270 30 9,785 15 9,673 75 2.10 11.34523 17,018 80 12,428 00 11,601 00 11,408 60 2.01 13.40364 19,474 90 14,558 80 13,675 80 13,470 70 2.02 15.19725 24,067 30 16,834 50 15,580 40 15,291 50 2.02 19.50933 27,595 60 21,173 20 19,900 80 19,604 10 2.04 19.73921 31,068 00 21,848 90 20,233 80 19,861 00 2.02 19.73921 31,068 00 21,848 90 20,233 80 19,861 00 2.02 D.o.f. 138 820 5544 40528 h 8.660 10−1 4.330 10−1 2.165 10−1 1.083 10−1 Prysmatic mesh

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 24 / 25

SLIDE 57

Discrete problems: numerics

Maxwell eigenvalues in 2D

Find ξh ∈ C 1, ξh = 0, λh ∈ R: [ a−1CURL ξh, CURL ξt

h]2 − λh[ξh, ξt h]1 = 0

∀ ξt

h

Exact Computed Slope

ref. =1

2 3 4 h 8.660 10− 4.330 10− 2.165 10− 1.083 10− Prysmatic mesh 9.639724 14,399 60 11,083 80 10,057 40 9,770 14 1.68 11.34523 15,982 90 12,406 10 11,613 10 11,415 60 1.93 13.40364 17,232 30 14,508 70 13,669 70 13,469 60 2.01 15.19725 18,969 10 18,577 80 15,918 50 15,360 90 2.14 19.50933 19,907 90 20,858 40 20,069 90 19,719 10 1.42 19.73921 22,073 60 21,329 10 20,145 00 19,826 20 2.22 19.73921 22,566 30 21,973 40 20,214 20 19,842 70 2.20 D.o.f. 340 1760 10720 73280 h 8.975 10−1 4.488 10−1 2.244 10−1 1.122 10−1

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 24 / 25

SLIDE 58

Open problems - conclusions

Spectral properties (eigenvalue computations).. theory? Use of these techniques for mass lumping for edge and face elements Anisotropic meshes: edge and face elements might work with “good“ choices of scalar products Construction of ⋆−Hodge operators on general meshes and space dimensions without sub-meshing High order mimetic finite differences

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 25 / 25

SLIDE 59

Open problems - conclusions

Spectral properties (eigenvalue computations).. theory? Use of these techniques for mass lumping for edge and face elements Anisotropic meshes: edge and face elements might work with “good“ choices of scalar products Construction of ⋆−Hodge operators on general meshes and space dimensions without sub-meshing High order mimetic finite differences Virtual version of the cochain complex ...? Would it be possible to obtain “regular” virtual elements? Thanks for the attention! http://www.imati.cnr.it/annalisa

A. Buffa (IMATI-CNR Italy)

Discrete Differential Forms 25 / 25