Local coderivatives and approximation of Hodge Laplace problems - - PowerPoint PPT Presentation

Local coderivatives and approximation of Hodge Laplace problems

Ragnar Winther

Department of Mathematics University of Oslo Norway

based on joint work with Jeonghun J. Lee, UT, Austin

1

The mixed finite element method

Elliptic problem: − div(K grad u) = f in Ω, u|∂Ω = 0 The mixed method:

• K −1σh, τ
• − uh, div τ = 0,

τ ∈ Σh ⊂ H(div), div σh, v = f , v , v ∈ Vh ⊂ L2, (1) where σh approximates −K grad u.

2

The mixed finite element method

Elliptic problem: − div(K grad u) = f in Ω, u|∂Ω = 0 The mixed method:

• K −1σh, τ
• − uh, div τ = 0,

τ ∈ Σh ⊂ H(div), div σh, v = f , v , v ∈ Vh ⊂ L2, (1) where σh approximates −K grad u. In contrast to the standard finite element method the mixed method is a conservative discretization.

2

The mixed finite element method

Elliptic problem: − div(K grad u) = f in Ω, u|∂Ω = 0 The mixed method:

• K −1σh, τ
• − uh, div τ = 0,

τ ∈ Σh ⊂ H(div), div σh, v = f , v , v ∈ Vh ⊂ L2, (1) where σh approximates −K grad u. In contrast to the standard finite element method the mixed method is a conservative discretization. However, the method is nonlocal.

2

Properties

Conservative:

• T

f dx =

• ∂T

σh · ν ds.

3

Properties

Conservative:

• T

f dx =

• ∂T

σh · ν ds. Nonlocal: The map uh → σh, which approximates u → −K grad u, defined by

• K −1σh, τ
• = uh, div τ = 0,

τ ∈ Σh is nonlocal.

3

Construction of local methods

◮ two point flux methods based on lumping of the

Raviart-Thomas method, Baranger, Maitre, Oudin (1993),

◮ multipoint flux approximations, Aavatsmark et al., (1998, . . .) ◮ various other finite volume schemes, mimetic methods ◮ discrete exterior calculus, Desbrun, Hirani, Leok, Marsden

(2005)

◮ lumping of the P1 − P0 method, Brezzi, Fortin, Marini (2006)

4

Multipoint flux approximations

The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method.

5

Multipoint flux approximations

The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method.

◮ Wheeler and Yotov, A multipoint flux mixed finite element... ,

SIAM J. Num. Anal. (2006)

◮ Droniou and Eymard, A mixed finite volume scheme.... on any

grid, Numer. Math. (2006)

◮ Klausen and W, Numer. Math. and Num. Meth. for PDE

(2006)

◮ Bause, Hoffman, Knabner, Multipoint flux approximations on

triangular grids, Numer. Math. (2010)

◮ Ingram, Wheeler, Yotov, A multipoint flux mixed finite

element method on hexahedra, SIAM J. Num. Anal. (2010)

◮ Wheeler and Yotov, A multipoint flux mixed finite element

method on distorted quadrilaterals hexahedra, Numer. Math. (2012)

5

Multipoint flux approximations

The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method.

◮ Wheeler and Yotov, A multipoint flux mixed finite element... ,

SIAM J. Num. Anal. (2006)

◮ Droniou and Eymard, A mixed finite volume scheme.... on any

grid, Numer. Math. (2006)

◮ Klausen and W, Numer. Math. and Num. Meth. for PDE

(2006)

◮ Bause, Hoffman, Knabner, Multipoint flux approximations on

triangular grids, Numer. Math. (2010)

◮ Ingram, Wheeler, Yotov, A multipoint flux mixed finite

element method on hexahedra, SIAM J. Num. Anal. (2010)

◮ Wheeler and Yotov, A multipoint flux mixed finite element

method on distorted quadrilaterals hexahedra, Numer. Math. (2012) in addition to the paper by Brezzi et al. from 2006.

5

Hodge Laplace problems and their discretizations

The de Rham complex: HΛ0(Ω)

d

− → HΛ1(Ω)

d

− → · · ·

d

− → HΛn(Ω)   π0

h

  π1

h

  πn

h

V 0

h d

− → V 1

h d

− → · · ·

d

− → V n

h

where HΛk(Ω) = {v ∈ L2Λk+1(Ω) | dv ∈ L2Λk(Ω)}, and V k

h is a subspace.

6

Hodge Laplace problems and their discretizations

The de Rham complex: HΛ0(Ω)

d

− → HΛ1(Ω)

d

− → · · ·

d

− → HΛn(Ω)   π0

h

  π1

h

  πn

h

V 0

h d

− → V 1

h d

− → · · ·

d

− → V n

h

where HΛk(Ω) = {v ∈ L2Λk+1(Ω) | dv ∈ L2Λk(Ω)}, and V k

h is a subspace.

We have a stable mixed methods for the Hodge Laplace problems iff there exists bounded cochain projections πh.

6

Mixed formulation of Hodge Laplace problems

The non-mixed formulation: Lu = (d∗d + dd∗)u = f , where the unknown u is a k-form. The operator d∗ mapping k-forms to (k − 1)-forms can be represented in two ways, either as the L2 adjoint of d, or as ⋆d⋆, where ⋆ is Hodge star operator mapping k-forms to n − k-forms.

7

Mixed formulation of Hodge Laplace problems

The non-mixed formulation: Lu = (d∗d + dd∗)u = f , where the unknown u is a k-form. The operator d∗ mapping k-forms to (k − 1)-forms can be represented in two ways, either as the L2 adjoint of d, or as ⋆d⋆, where ⋆ is Hodge star operator mapping k-forms to n − k-forms. The mixed formulation: Find (σ, u) ∈ HΛk−1(Ω) × HΛk(Ω) such that σ, τ − u, dτ = 0, τ ∈ HΛk−1(Ω), dσ, v + du, dv = f , v , v ∈ HΛk(Ω), where σ = d∗u, and ·, · denotes L2-inner products.

7

The corresponding mixed method

Find (σh, uh) ∈ V k−1

h

× V k

h such that

σh, τ − uh, dτ = 0, τ ∈ V k−1

h

, dσh, v + duh, dv = f , v , v ∈ V k

h ,

where the operator d∗

h : V k h → V k−1 h

defined by d∗

hu, τ = u, dτ = 0,

τ ∈ V k−1

h

is nonlocal.

8

The corresponding mixed method

Find (σh, uh) ∈ V k−1

h

× V k

h such that

σh, τ − uh, dτ = 0, τ ∈ V k−1

h

, dσh, v + duh, dv = f , v , v ∈ V k

h ,

where the operator d∗

h : V k h → V k−1 h

defined by d∗

hu, τ = u, dτ = 0,

τ ∈ V k−1

h

is nonlocal. In particular, there is no natural ”discrete Hodge star operator,” ⋆h, mapping V k

h to V n−k h

, cf. Hiptmair, Discrete Hodge operators, (2001).

8

Examples of Hodge Laplace problems

The case k = n corresponds to mixed formulation of the scalar Laplace problem with HΛn−1 × HΛn replaced by H(div) × L2, while the case k = 0 corresponds to the standard H1 formulation.

9

Examples of Hodge Laplace problems

The case k = n corresponds to mixed formulation of the scalar Laplace problem with HΛn−1 × HΛn replaced by H(div) × L2, while the case k = 0 corresponds to the standard H1 formulation. For n = 3 and k = 1: Find (σ, u) ∈ H1 × H(curl) such that σ, τ − u, grad τ = 0, τ ∈ H1, grad σ, v + curl u, curl v = f , v , v ∈ H(curl). For n = 3 and k = 2: Find (σ, u) ∈ H(curl) × H(div) such that σ, τ − u, curl τ = 0, τ ∈ H(curl), curl σ, v + div u, div v = f , v , v ∈ H(div). Both correponds to vector Laplace.

9

Abstract framework, Hilbert complexes

• cf. Arnold, Falk, W, Bulletin of AMS, 2010.

10

Abstract framework, Hilbert complexes

• cf. Arnold, Falk, W, Bulletin of AMS, 2010.

We are given a structure of the form V 0

d

− → V 1

d

− → · · ·

d

− → V n   π0

h

  π1

h

  πn

h

V 0

h d

− → V 1

h d

− → · · ·

d

− → V n

h

where V k = {v ∈ W k | dv ∈ W k+1}, and V k

h ⊂ V k.

10

Abstract framework, Hilbert complexes

• cf. Arnold, Falk, W, Bulletin of AMS, 2010.

We are given a structure of the form V 0

d

− → V 1

d

− → · · ·

d

− → V n   π0

h

  π1

h

  πn

h

V 0

h d

− → V 1

h d

− → · · ·

d

− → V n

h

where V k = {v ∈ W k | dv ∈ W k+1}, and V k

h ⊂ V k.

The standard mixed method for the Hodge Laplace problems: Find (˜ σh, ˜ uh) ∈ V k−1

h

× V k

h such that

˜ σh, τ − dτ, ˜ uh = 0, τ ∈ V k−1

h

, d˜ σh, v + d ˜ uh, dv = f , v , v ∈ V k

h .

where ·, · are inner products of the W spaces.

10

Abstract framework, Hilbert complexes

• cf. Arnold, Falk, W, Bulletin of AMS, 2010.

We are given a structure of the form V 0

d

− → V 1

d

− → · · ·

d

− → V n   π0

h

  π1

h

  πn

h

V 0

h d

− → V 1

h d

− → · · ·

d

− → V n

h

where V k = {v ∈ W k | dv ∈ W k+1}, and V k

h ⊂ V k.

The standard mixed method for the Hodge Laplace problems: Find (˜ σh, ˜ uh) ∈ V k−1

h

× V k

h such that

˜ σh, τ − dτ, ˜ uh = 0, τ ∈ V k−1

h

, d˜ σh, v + d ˜ uh, dv = f , v , v ∈ V k

h .

where ·, · are inner products of the W spaces. The standard theory gives optimal convergence in the V -norms for stable methods.

10

Variational crime

Find (σh, uh) ∈ V k−1

h

× V k

h such that

σh, τh − dτ, uh = 0, τ ∈ V k−1

h

, dσh, v + duh, dv = f , v , v ∈ V k

h ,

where ·, ·h is an alternative inner product on V k−1

h

.

11

Variational crime

Find (σh, uh) ∈ V k−1

h

× V k

h such that

σh, τh − dτ, uh = 0, τ ∈ V k−1

h

, dσh, v + duh, dv = f , v , v ∈ V k

h ,

where ·, ·h is an alternative inner product on V k−1

h

. We will give two conditions such that stability and convergence of the perturbed method is maintained.

11

Stability and consistency

Stability condition: The inner products ·, · and ·, ·h are equivalent on V k−1

h

, uniformly in h.

12

Stability and consistency

Stability condition: The inner products ·, · and ·, ·h are equivalent on V k−1

h

, uniformly in h. Consistency condition: There exist discrete subspaces W k−1

h

⊂ W k−1 and ˜ V k−1

h

⊂ V k−1

h

that (⋆) τ, τ0 = τ, τ0h , τ ∈ ˜ V k−1

h

, τ0 ∈ W k−1

h

, and a linear map Πh : V k−1

h

→ ˜ V k−1

h

such that dΠhτ = dτ, Πhτ τ, and Πhτ, τ0h = τ, τ0h , τ0 ∈ W k−1

h

.

12

Stability and consistency

Stability condition: The inner products ·, · and ·, ·h are equivalent on V k−1

h

, uniformly in h. Consistency condition: There exist discrete subspaces W k−1

h

⊂ W k−1 and ˜ V k−1

h

⊂ V k−1

h

that (⋆) τ, τ0 = τ, τ0h , τ ∈ ˜ V k−1

h

, τ0 ∈ W k−1

h

, and a linear map Πh : V k−1

h

→ ˜ V k−1

h

such that dΠhτ = dτ, Πhτ τ, and Πhτ, τ0h = τ, τ0h , τ0 ∈ W k−1

h

. If (⋆) holds with ˜ V k−1

h

= V k−1

h

, then the rest holds with Πh = I.

12

Convergence result

We have the following estimate for the perturbed method: σh − ˜ σhV + uh − ˜ uhV σ − PWhσ + σ − ˜ σh

13

Simplicial meshes

Standard families of spaces: R − → P−

r Λ0(Th) d

− → P−

r Λ1(Th) d

− → · · ·

d

− → P−

r Λn(Th) −

→ 0, R − → PrΛ0(Th)

d

− → Pr−1Λ1(Th)

d

− → · · ·

d

− → Pr−nΛn(Th) − → 0.

14

Simplicial meshes

Standard families of spaces: R − → P−

r Λ0(Th) d

− → P−

r Λ1(Th) d

− → · · ·

d

− → P−

r Λn(Th) −

→ 0, R − → PrΛ0(Th)

d

− → Pr−1Λ1(Th)

d

− → · · ·

d

− → Pr−nΛn(Th) − → 0. The simplest space is P−

1 Λk(Th), i.e. the space of Whitney forms,

and we have P−

1 Λk(Th) ⊂ PrΛk(Th) ⊂ P− r+1Λk(Th) ⊂ Pr+1Λk(Th),

r ≥ 1.

14

The P1Λk−1(Th) − P−

1 Λk(Th) method

We will consider: V k−1

h

= P1Λk−1(Th) V k

h = P− 1 Λk(Th)

which is a generalization of Brezzi, Fortin, Marini (2006). The ”coderivative” d∗

h is then defined by

d∗

hu, τh = u, dτ ,

u ∈ P−

1 Λk(Th), τ ∈ P1Λk−1(Th).

We need to define the inner product ·, ·h such it satisfies the stability and consistency condition, and such that the operator d∗

h : P− 1 Λk(Th) → P1Λk−1(Th) is local.

15

Degrees of freedom for the space P1Λk(Th)

The standard DOFs for P1Λk(Th) is

• f

trf u ∧ v, v ∈ P1Λ0(f ), f ∈ ∆k(Th), and dim P1Λk(Th) = (k + 1)|∆k(Th)|.

16

Degrees of freedom for the space P1Λk(Th)

The standard DOFs for P1Λk(Th) is

• f

trf u ∧ v, v ∈ P1Λ0(f ), f ∈ ∆k(Th), and dim P1Λk(Th) = (k + 1)|∆k(Th)|. Assume f = [x0, x1, . . . , xk] ∈ ∆k(Th). Define φf ,xi(u) = uxi(x0 − xi, . . . , xi−1 − xi, xi+1 − xi, . . . , xk − xi), for i = 0, 1, . . . , k.

16

Degrees of freedom for the space P1Λk(Th)

The standard DOFs for P1Λk(Th) is

• f

trf u ∧ v, v ∈ P1Λ0(f ), f ∈ ∆k(Th), and dim P1Λk(Th) = (k + 1)|∆k(Th)|. Assume f = [x0, x1, . . . , xk] ∈ ∆k(Th). Define φf ,xi(u) = uxi(x0 − xi, . . . , xi−1 − xi, xi+1 − xi, . . . , xk − xi), for i = 0, 1, . . . , k. Alternative DOFs: {φf ,x(u)} for all pairs (f , x), f ∈ ∆k(Th), x ∈ ∆0(f ).

16

Basis and for P1Λk(Th)

We let {ψf ,x} be the corresponding dual basis. Note that (ψf ,x)y ≡ 0 if y ∈ ∆0(Th) and y = x.

17

Basis and for P1Λk(Th)

We let {ψf ,x} be the corresponding dual basis. Note that (ψf ,x)y ≡ 0 if y ∈ ∆0(Th) and y = x. In fact: ψf ,xi = λidλ0 ∧ · · · ∧ dλi−1 ∧ dλi+1 ∧ · · · ∧ dλk, where f = [x0, x1, . . . xk] and λi the barycentric coordinate of xi.

17

Modified inner product on P1Λk(Th)

We recall that the standard inner product on L2Λk(Ω) can be expressed as u, v =

• Ω

ux, vxAlt dx.

18

Modified inner product on P1Λk(Th)

We recall that the standard inner product on L2Λk(Ω) can be expressed as u, v =

• Ω

ux, vxAlt dx. The modified inner product: u, vh =

• T∈Th

u, vh,T , u, vh,T =

n

• i=0

|T| n + 1 uxi, vxiAlt . where T = [x0, x1, . . . xn].

18

Locality and consistency

Since the the matrix ψf ,x, ψg,yh is block diagonal the operator d∗

h is local.

19

Locality and consistency

Since the the matrix ψf ,x, ψg,yh is block diagonal the operator d∗

h is local.

The consistency condition in this case ( ˜ V k−1

h

⊂ V k−1

h

): τ, τ0 = τ, τ0h , τ ∈ V k−1

h

, τ0 ∈ W k−1

h

, where W k−1

h

:= {τ ∈ L2Λk−1(Ω) | τ|T ∈ P0Λk−1(T) for all T ∈ Th}.

19

Locality and consistency

Since the the matrix ψf ,x, ψg,yh is block diagonal the operator d∗

h is local.

The consistency condition in this case ( ˜ V k−1

h

⊂ V k−1

h

): τ, τ0 = τ, τ0h , τ ∈ V k−1

h

, τ0 ∈ W k−1

h

, where W k−1

h

:= {τ ∈ L2Λk−1(Ω) | τ|T ∈ P0Λk−1(T) for all T ∈ Th}. The extra error introduced by the perturbation is essential bounded by σ − PWhσ.

19

An explicit formula for the elements of the mass matrix

We have ψf ,x0, ψg,x0h =

• T

|T| n + 1 (ψf ,x0)x0, (ψg,x0)x0Alt = 1 (n + 1)(k!)2

• T

|T| |f | |g| volf , volgAlt , where the sum is over all T ∈ Th such that both f and g are in ∆k(T).

20

Cubical meshes

Standard families of spaces according to Arnold and Awanou (2014) and Arnold, Boffi, Bonizzoni (2015): R − → Q−

r Λ0(Th) d

− → Q−

r Λ1(Th) d

− → · · ·

d

− → Q−

r Λn(Th) −

→ 0, R − → SrΛ0(Th)

d

− → Sr−1Λ1(Th)

d

− → · · ·

d

− → Sr−nΛn(Th) − → 0.

21

Cubical meshes

Standard families of spaces according to Arnold and Awanou (2014) and Arnold, Boffi, Bonizzoni (2015): R − → Q−

r Λ0(Th) d

− → Q−

r Λ1(Th) d

− → · · ·

d

− → Q−

r Λn(Th) −

→ 0, R − → SrΛ0(Th)

d

− → Sr−1Λ1(Th)

d

− → · · ·

d

− → Sr−nΛn(Th) − → 0. The simplest space is Q−

1 Λk(Th), i.e.

Q−

1 Λk(Th) ⊂ Q− r Λk(Th), SrΛk(Th)

and as the Whitney forms this space has one DOF for each k dimensional subcomplex of Th.

21

The space Q−

1 Λk(Th)

Local shape functions on T ∈ Th: u|T =

• σ∈Σ(k)

(

• αj≤1−δj,σ

cαxα)dxσ, where dxσ = dxσ1 ∧ · · · ∧ dxσk. The local space Q−

1 Λk(T) has dimension 2n−k

n k

• .

22

The space Q−

1 Λk(Th)

Local shape functions on T ∈ Th: u|T =

• σ∈Σ(k)

(

• αj≤1−δj,σ

cαxα)dxσ, where dxσ = dxσ1 ∧ · · · ∧ dxσk. The local space Q−

1 Λk(T) has dimension 2n−k

n k

• .

Degrees of freedom:

• f

trf u, f ∈ ∆k(Th), and dim Q−

1 Λk(Th) = |∆k(Th)|.

22

The space S1Λk(Th)

Local shape functions on T ∈ Th: u|T = u− + dκ

• σ∈Σ(k)
• i∈σ

(

• αj≤1−δj,σ

cαxα)xi dxσ, T ∈ Th, where u− ∈ Q−

1 Λk(T) and where κ is the Koszul operator given by

(κu)x = uxx. Alternatively, κ(dxσ) = κ(dxσ1∧· · ·∧dxσk) =

k

• i=1

(−1)i+1xσidxσ1∧· · ·∧ dxσi∧· · ·∧dxσk,

23

The space S1Λk(Th)

Local shape functions on T ∈ Th: u|T = u− + dκ

• σ∈Σ(k)
• i∈σ

(

• αj≤1−δj,σ

cαxα)xi dxσ, T ∈ Th, where u− ∈ Q−

1 Λk(T) and where κ is the Koszul operator given by

(κu)x = uxx. Alternatively, κ(dxσ) = κ(dxσ1∧· · ·∧dxσk) =

k

• i=1

(−1)i+1xσidxσ1∧· · ·∧ dxσi∧· · ·∧dxσk, The dimension of the local space S1Λk(T) is 2n−k n k

• (k + 1).

23

The space S1Λk(Th)

Local shape functions on T ∈ Th: u|T = u− + dκ

• σ∈Σ(k)
• i∈σ

(

• αj≤1−δj,σ

cαxα)xi dxσ, T ∈ Th, where u− ∈ Q−

1 Λk(T) and where κ is the Koszul operator given by

(κu)x = uxx. Alternatively, κ(dxσ) = κ(dxσ1∧· · ·∧dxσk) =

k

• i=1

(−1)i+1xσidxσ1∧· · ·∧ dxσi∧· · ·∧dxσk, The dimension of the local space S1Λk(T) is 2n−k n k

• (k + 1).

Degrees of freedom:

• f

trf u ∧ v, v ∈ P1Λ0(f ), f ∈ ∆k(Th). and dim S1Λk(Th) = (k + 1)|∆k(Th)|.

23

The local space S+

1 Λk(T)

is defined by S+

1 Λk(T) = Q− 1 Λk(T) + dκBΛk(T).

Here BΛk(T) = span{pσ∗pσdxσ | σ ∈ Σ(k) }, where pσ ∈ Q1(Rk) and pσ∗ ∈ Q1(Rn−k) are polynomials in the variables {xj}j∈σ and {xj}j∈σ∗, respectively, and where pσ(0) = 0.

24

The local space S+

1 Λk(T)

is defined by S+

1 Λk(T) = Q− 1 Λk(T) + dκBΛk(T).

Here BΛk(T) = span{pσ∗pσdxσ | σ ∈ Σ(k) }, where pσ ∈ Q1(Rk) and pσ∗ ∈ Q1(Rn−k) are polynomials in the variables {xj}j∈σ and {xj}j∈σ∗, respectively, and where pσ(0) = 0. So Q−

1 Λk(T) ⊂ S1Λk(T) ⊂ S+ 1 Λk(T),

and Q1Λk(T) = Q−

1 Λk(T) ⊕ BΛk(T).

and dim S+

1 Λk(T) = dim Q1Λk(T) = 2n

n k

• .

24

DOFs for S+

1 Λk(Th)

The natural set of DOFs are

• f

trf u ∧ v, v ∈ Q1Λ0(f ), f ∈ ∆k(T), and dim S+

1 Λk(Th) = 2k|∆k(Th)|.

25

Altrnative DOFs for S+

1 Λk(Th)

For each f ∈ ∆k(Th) and each x0 ∈ ∆0(f ) define φf ,x0(u) = ux0(x1 − x0, x2 − x0, . . . , xk − x0), where {xj}k

j=1 are the k vertices of f such that [x0, xj] ∈ ∆1(f ).

The functionals {φf ,x}, where f ∈ ∆k(Th) and x ∈ ∆0(f ) is an alternative set of DOFs.

26

Altrnative DOFs for S+

1 Λk(Th)

For each f ∈ ∆k(Th) and each x0 ∈ ∆0(f ) define φf ,x0(u) = ux0(x1 − x0, x2 − x0, . . . , xk − x0), where {xj}k

j=1 are the k vertices of f such that [x0, xj] ∈ ∆1(f ).

The functionals {φf ,x}, where f ∈ ∆k(Th) and x ∈ ∆0(f ) is an alternative set of DOFs. We let {ψf ,x} be the corresponding dual basis, and the modified inner product is defined by u, vh =

• T∈Th

u, vh,T , where u, vh,T = 2−n|T|

• x∈∆0(T)

ux, vxAlt .

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Altrnative DOFs for S+

1 Λk(Th)

For each f ∈ ∆k(Th) and each x0 ∈ ∆0(f ) define φf ,x0(u) = ux0(x1 − x0, x2 − x0, . . . , xk − x0), where {xj}k

j=1 are the k vertices of f such that [x0, xj] ∈ ∆1(f ).

The functionals {φf ,x}, where f ∈ ∆k(Th) and x ∈ ∆0(f ) is an alternative set of DOFs. We let {ψf ,x} be the corresponding dual basis, and the modified inner product is defined by u, vh =

• T∈Th

u, vh,T , where u, vh,T = 2−n|T|

• x∈∆0(T)

ux, vxAlt . The matrix ψf ,x, ψg,yh is block diagonal.

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The local method

We use V k−1

h

= S+

1 Λk(Th) and V k h = Q− 1 Λk(Th) and the inner

pruct ·, ·h. This gives a local d∗

h operator.

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The local method

We use V k−1

h

= S+

1 Λk(Th) and V k h = Q− 1 Λk(Th) and the inner

pruct ·, ·h. This gives a local d∗

h operator.

The consistency condition: τ, τ0 = τ, τ0h , τ ∈ ˜ V k−1

h

, τ0 ∈ W k−1

h

, and there is a linear map Πh : V k−1

h

→ ˜ V k−1

h

such that dΠhτ = dτ, and Πhτ, τ0h = τ, τ0h , τ0 ∈ W k−1

h

holds with W k−1

h

equal piecewise constants, ˜ V k−1

h

= Q−

1 Λk−1(Th),

and Πh the canonical projection defined by the degrees of freedom

• f Q−

1 Λk−1(Th).

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Conclusion

◮ By perturbing the standard mixed methods of the Hodge

Laplace problems we have constructed stable, convergent and local approximations.

◮ The method are of low order and can also be seen as finite

difference methods.

◮ The approach leads to a natural path to convergence

estimates based on standard finite element theory and variational crimes.

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