Mimetic Least Squares Spectral/ hp Finite Element Method for the - - PowerPoint PPT Presentation
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Mimetic Least Squares Spectral/ hp Finite Element Method for the Poisson Equation Artur Palha 1 and Marc Gerritsma 1 1 Faculty of Aerospace Engineering Delft University
Outline The Standard Least Squares Mimetic Approach Summary and Future Work
1Faculty of Aerospace Engineering
Delft University of Technology Email: a.palhadasilvaclerigo@tudelft.nl
Artur Palha and Marc Gerritsma Mimetic Least Squares 1 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work
Artur Palha and Marc Gerritsma Mimetic Least Squares 2 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
Artur Palha and Marc Gerritsma Mimetic Least Squares 3 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
= f in Ω Ru = h
Γ
Artur Palha and Marc Gerritsma Mimetic Least Squares 4 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
= f in Ω Ru = h
Γ
= f in Ω Run,p = h
Γ
Artur Palha and Marc Gerritsma Mimetic Least Squares 4 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
min
uh,p∈Xh,p
I(uh,p; f, h) ≡ 1 2
Xh,Ω + Ruh,p − h2 Xh,Ω
Artur Palha and Marc Gerritsma Mimetic Least Squares 5 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
min
uh,p∈Xh,p
I(uh,p; f, h) ≡ 1 2
Xh,Ω + Ruh,p − h2 Xh,Ω
Auh,p = b
Artur Palha and Marc Gerritsma Mimetic Least Squares 5 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
φ(x, y) → φh(x, y) =
φi,jhp
i (x)hp j (y)
u(x, y) → uh(x, y) =
m,n ux m,nhp m(x)hp n(y)
m,nhp k(x)hp l (y)
φh(x, y) ∈ span
i (x)hp j (y)
i, j = 0, . . . , p uh(x, y) ∈ span
m(x)hp n(y) ⊗ hp k(x)hp l (y)
m, n, k, l = 1, . . . , p hp
i (ξ) Lagrange interpolants over Gauss-Lobatto-Legendre points. Artur Palha and Marc Gerritsma Mimetic Least Squares 6 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
mimetic φ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 0.0 0.3 0.6 0.9 1.2 1.5 1.8 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 0.0 0.3 0.6 0.9 1.2 1.5 1.8
Artur Palha and Marc Gerritsma Mimetic Least Squares 7 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
mimetic vx mimetic qx
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6
Artur Palha and Marc Gerritsma Mimetic Least Squares 8 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
mimetic vy mimetic qy
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.20 1.54 0.88 0.22 0.44 1.10 1.76
Artur Palha and Marc Gerritsma Mimetic Least Squares 9 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work?
Artur Palha and Marc Gerritsma Mimetic Least Squares 10 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Artur Palha and Marc Gerritsma Mimetic Least Squares 11 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
There is an intrinsic association between physical quantities and geometrical objects: ◮ Points: e.g. Electric potential, φ ◮ Lines: e.g. Electric field, E, Magnetizing field, H ◮ Surfaces: e.g. Magnetic flux, B, Electric displacement field, D ◮ Volumes: e.g. Charge density, ρ These associations are intrinsic to the differential equations that relate the physical quantities: ∇ · D = ρ ∇ · B = ∇ × E = − ∂B
∂t
∇ × H = J + ∂D
∂t
D = ǫE B = µH ⇐ ⇒
= Q(V )
=
= − ∂
∂t
=
∂t
D = ǫE B = µH
Artur Palha and Marc Gerritsma Mimetic Least Squares 12 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Configuration variables: variables that give the configuration of a physical system. How the system is described. (electric potential V , electric field vector E, velocity vector v). Associated to inner oriented manifolds Source variables: variables that describe the sources of the field or the forces acting
q). Associated to outer oriented manifolds Energy variables: variables obtained as the product of a configuration variable with a source variable.
Artur Palha and Marc Gerritsma Mimetic Least Squares 13 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Artur Palha and Marc Gerritsma Mimetic Least Squares 14 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Are characterized by the fact that their validity is independent of the nature of the medium under consideration. Connect configuration variables with configuration variables and source variables with source variables. Are independent of metric since they are intrinsically integral equations (global). ∇ · B = 0, ∇φ = u, ∇ × E = 0, . . .
Are characterized by the fact that their validity depends on the nature of the medium under consideration. They describe the behaviour of a material. Connect configuration variables with source variables. Depend on the metric since they are intrinsically local in nature. D = ǫE, q = ρv, . . .
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
∇φ = v ∇ · q = f q = ρv ◮ There is no reference to which geometrical object the physical quantities are associated. ◮ There is no reference to inner or outer orientation. ◮ All this is given a posteriori. Right hand rule and so on. How to solve this?
Artur Palha and Marc Gerritsma Mimetic Least Squares 16 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
∇φ = v ∇ · q = f q = ρv ◮ There is no reference to which geometrical object the physical quantities are associated. ◮ There is no reference to inner or outer orientation. ◮ All this is given a posteriori. Right hand rule and so on. How to solve this?
Artur Palha and Marc Gerritsma Mimetic Least Squares 16 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Artur Palha and Marc Gerritsma Mimetic Least Squares 17 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
We need to introduce 1 object and 4 operators: ◮ k-differential form or k-form ◮ wedge product, ∧ ◮ inner product, (·, ·) ◮ exterior derivative, d ◮ Hodge-⋆ operator, ⋆
Artur Palha and Marc Gerritsma Mimetic Least Squares 18 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
A k-differential form, or k-form, is a k-linear and antisymmetric mapping: ωk : TMx × . . . × TMx
→ F Where TMx is the tangent space of a k-differentiable manifold (smooth k-dimensional surface) at x and F is a field, which in this paper is R.
Can be seen as a machine that eats k-vectors and spits a number. Under integration can be seen as a machine that eats k-manifolds and returns a number. Intrinsically connected to geometry and integration.
Artur Palha and Marc Gerritsma Mimetic Least Squares 19 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
The wedge product, ∧, in a n-dimensional space, is defined as a mapping: ∧ : Λk × Λl → Λk+l, k + l < n, with the property that: ωk ∧ αl = (−1)klαl ∧ ωk, where Λk is the space of k-forms.
Can be seen as a machine that eats a k-form and a l-form and returns a (k + l)-form. Or simply as the generalization of the exterior product to arbitrary dimensions.
Artur Palha and Marc Gerritsma Mimetic Least Squares 20 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
This means that, in Rn, one can write any k-form as: ωk =
wi1,...,ik(x)dxi1 ∧ . . . ∧ dxik, where dxi are the orthonormal basis 1-forms. In this way one can define the set of functions
correspondence between these sets and k-forms. Moreover, one can verify that these sets, in R2 for example, have a correspondence to well known entities: scalar fields as proxies for 0-forms and 2-forms, vector fields as proxies for 1-forms.
Artur Palha and Marc Gerritsma Mimetic Least Squares 21 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
The well known inner product in Rn and the correspondence between k-forms and their proxies induces the definition of an inner product for k-forms as a mapping (·, ·) : Λk × Λk → R: (αk, βk) =
ai1,...,ik(x)bi1,...,ik(x) An L2 inner product of k-forms, in a n-dimensional space, can also be defined as a mapping:
Ω =
ωn Where we have introduced the volume n-form ωn, on an n-dimensional space.
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
The exterior derivative d, in a n-dimensional space, is a mapping: d : Λk → Λk+1, k = 0, 1, . . . , n − 1, which satisfies: d
= dωk ∧ αl + (−1)kωk ∧ dαl, k + l < n and ddωk = 0, ∀ω ∈ Λk, k < n − 1
It is simply a generalization of the well known differential operators, ∇, ∇× and ∇·, to higher dimensions. In R3: dφ0 = ∇φ, dv1 = ∇ × v, de2 = ∇ ·
Artur Palha and Marc Gerritsma Mimetic Least Squares 23 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
The Hodge-⋆ operator, in an n-dimensional space, is a mapping: ⋆ : Λk → Λn−k, k ≤ n, that satisfies: α ∧ ⋆β = (α, β) ωn, ∀α, β ∈ Λk.
A machine that transforms k-forms into twisted (n − k)-forms. Changes orientation from inner to outer and vice-versa.
Artur Palha and Marc Gerritsma Mimetic Least Squares 24 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Maxwell equations: dF 2 = 0, dG2 = J3, dJ3 = 0, G2 = ⋆F 2 Fundamental theorems:
dωk =
ωk
Connection between k-forms and k + 1-manifolds:
d2 = ⋆ǫe1, q2 = ⋆ρv1, . . .
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Additionally, one verifies that in sufficiently regular regions Ω the spaces of differential forms together with the exterior derivative d constitute an exact sequence, called the de Rham complex, which, in 3D is: R
Λ0
⋆
Λ1
⋆
Λ2
⋆
Λ3
⋆
Λ3
Λ2
d
Λ1
d
Λ0
d
Mimetic Least Squares 26 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
And in 2D reduces to: R
Λ0
⋆
Λ1
⋆
Λ2
⋆
Λ2
Λ1
d
Λ0
d
more
R
H1
⋆
⋆
L2
⋆
H1(div)
∇·
∇⊥
Mimetic Least Squares 27 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
∇φ = v ∇ · q = f q = ρv ⇔ dφ0 = v1 dq1 = f2 q1 = ⋆ρv1
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
∇φ = v ∇ · q = f q = ρv ⇔ dφ0 = v1 dq1 = f2 q1 = ⋆ρv1
φ0
d
u1
⋆
v1
d
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
◮ Depends on the existence of a pair of topologically dual grids ◮ One-to-one correspondence between dual variables ◮ Simple Hodge-⋆ operator ◮ All equations satisfied globally
◮ Sacrifices one of the equillibrium equations ◮ Satisfies exactly the other equilibrium equation and the constitutive equation locally
◮ Satisfies exactly the equilibrium equations ◮ Relaxes the constitutive equation, being enforced weakly
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
◮ Depends on the existence of a pair of topologically dual grids ◮ One-to-one correspondence between dual variables ◮ Simple Hodge-⋆ operator ◮ All equations satisfied globally
◮ Sacrifices one of the equillibrium equations ◮ Satisfies exactly the other equilibrium equation and the constitutive equation locally
◮ Satisfies exactly the equilibrium equations ◮ Relaxes the constitutive equation, being enforced weakly
Artur Palha and Marc Gerritsma Mimetic Least Squares 29 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
◮ Impose the constitutive equation weakly ◮ Hodge-⋆ operator defined implicitly ◮ Minimize local discrepancy between dual variables
Seek (φ0
h, v1 h, q1 h) in Λ0 h × Λ1 h × Λ1 h such that
(1) I(φ0
h, v1 h, q1 h) = 1 2
h + v1 h2 0 + dq1 h − f22
dφ0
h = v1 h Artur Palha and Marc Gerritsma Mimetic Least Squares 30 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
If the subspaces Λ0
h, Λ1 h and Λ2 h are chosen in such a way that they constitute a de
Rham complex: R → Λ0
h d
− → Λ1
h d
− → Λ2
h → 0
then dφ0
h = v1 h is satisfied exactly. The problem becomes:
Seek (φ0
h, q1 h) in Λ0 h × Λ1 h such that
(2) I(φ0
h, q1 h) = 1 2
h + dφ02 0 + dq1 h − f22
different dual spaces.
Artur Palha and Marc Gerritsma Mimetic Least Squares 31 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
Find adequate subspaces Λ0
h, Λ1 h and Λ2 h must be specified. Since one will use a
spectral/hp LS method, these spaces are defined as: Λ0
h,p = span
i (x)hp j (y)
i = 0, . . . , p j = 0, . . . , p Λ1
h,p = span
hp−1
i
(x)hp
j (y) ⊗ hp n(x)˜
hp−1
m
(y)
i, m = 1, . . . , p j, n = 0, . . . , p Λ2
h,p = span
hp−1
i
(x)˜ hp−1
j
(y)
i = 1, . . . , p j = 1, . . . , p ◮ hp
i (ξ): i-th Lagrange interpolant of order p throught Gauss-Lobatto-Legendre
points ◮ ˜ hp
i (ξ): i-th Lagrange interpolant of order p throught Gauss points
◮ Degrees of freedom are located where they should be: at nodal points (for 0-forms), at edges (for 1-forms) and at volumes (for 2-forms). ◮ Different continuity properties ◮ These subspaces constitute a de Rham complex
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
standard φ
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 0.0 0.3 0.6 0.9 1.2 1.5 1.8 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 0.0 0.3 0.6 0.9 1.2 1.5 1.8
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
standard vx
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6
Artur Palha and Marc Gerritsma Mimetic Least Squares 34 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
standard vx
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6
Artur Palha and Marc Gerritsma Mimetic Least Squares 35 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
standard vy
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6
Artur Palha and Marc Gerritsma Mimetic Least Squares 36 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
standard vy
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 1.4 0.8 0.2 0.4 1.0 1.6
Artur Palha and Marc Gerritsma Mimetic Least Squares 37 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
figures/plots/convergence/convergence_phi_u.pdf figures/plots/convergence/convergence_phi_u.pdf
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Going back to the basics Differential geometry Mimetic least-squares
2 4 6 8 10 12 p 10-4 10-3 10-2 10-1 100 101 102
ǫ
conservative L2 error - φ standard L2 error - φ conservative L2 error - q standard L2 error - q 2 4 6 8 10 12 p 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
∇ ×q/∇ ·q
∇ ×q conservative ∇ ×q standard ∇ ·q conservative ∇ ·q standard
For Histopolants see: Robidoux, Polynomial Histopolation, Superconvergent Degrees Of Freedom, And Pseudospectral Discrete Hodge Operators, to appear.
Artur Palha and Marc Gerritsma Mimetic Least Squares 39 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
◮ Physical quantities are inherently geometrical ◮ Structure of PDE’s must be obeyed ◮ There is more to life than scalars and vectors
Artur Palha and Marc Gerritsma Mimetic Least Squares 40 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
◮ Curved domains ◮ 3 dimensions ◮ 4 dimensions: space-time
Artur Palha and Marc Gerritsma Mimetic Least Squares 41 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
Artur Palha and Marc Gerritsma Mimetic Least Squares 42 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
Tonti, E.: On the formal structure of physical theories. Consiglio Nazionale delle Ricerche, Milano (1975) Bochev, P. and Hyman, J.: Principles of mimetic discretizations of differential
Mattiussi, C.: An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comp. Physics 133, 289–309 (1997) Desbrun, M. and Kanso, E. and Tong, Y.: Discrete differential forms for computational modeling. SIGGRAPH ’05: ACM SIGGRAPH 2005 Courses (2005) Bossavit, A.: On the geometry of electromagnetism. J. Japan Soc. Appl.
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Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
Usually the 2-dimensional case is viewed as a special case and hence it is expressed by two exact sequences: R ֒ → H1
∇
− − → H1(curl)
∇×
− − − → L2 (3) R ֒ → H1
∇⊥
− − − → H1(div)
∇·
− − → L2 with ∇φ = ∂φ ∂x ex + ∂φ ∂y ey (4) ∇⊥φ = − ∂φ ∂y ex + ∂φ ∂x ey (5) ∇ × W = ∂Wy ∂x ex − ∂Wx ∂y ey (6) ∇ · W = ∂Wx ∂x + ∂Wy ∂y
back Artur Palha and Marc Gerritsma Mimetic Least Squares 44 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
These two exact sequences are obtained by a restriction of the 3D exact sequence: R ֒ → H1
∇
− − → H1(curl)
∇×
− − − → H1(div)
∇·
− − → L2 to a planar 2D surface embedded in R3, for example the xy-plane. This, in turn, reduces to the pair of exact sequences Eq. (3) and Eq. (??), since the vectors of the form φez can be identified with scalar functions. The odd operator ∇⊥ is, then, nothing but the result of applying the 3D ∇× to φez. The full de Rham complex in 2D becomes: R
H1
⋆
⋆
L2
⋆
H1(div)
∇·
∇⊥
Artur Palha and Marc Gerritsma Mimetic Least Squares 45 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
It is important to realize that this full exact complex is always relative to the proxies of the differential forms, that is, scalar and vector fields, not the differential k-forms to which they are associated. This is the important point, since this is the reason why both De Rham complexes are equivalent. Let us therefore show how the differential formulation agrees with the vector formulation with its special characteristics in 2D: R
Λ0
⋆
Λ1
⋆
Λ2
⋆
Λ2 ˜ Λ1
d
Λ0
d
Λ0, ˜ Λ1 and ˜ Λ2 are the spaces of twisted forms (as in Burke (1985) section 28,
back Artur Palha and Marc Gerritsma Mimetic Least Squares 46 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
To show that both exact complexes are equivalent one must show how to pass from forms to proxies and from proxies to forms. This is done by using the sharp, ♯, and flat, ♭, operators, respectively. The sharp, ♯, and flat, ♭, acting on a 0-form (φ0) and a scalar field (φ), give:
φ♭ = φ0 The sharp, ♯, and flat, ♭, acting on a 1-form (α1 = fdx + gdy) and a vector field (A = fex + gey), give:
A♭ = fdx + gdy The sharp, ♯, and flat, ♭, acting on a twisted 1-form (˜ β1 = −gdx + fdy) and a vector field (B = fex + gey):
β1♯ = fex + gey, B♭ = −gdx + fdy
back Artur Palha and Marc Gerritsma Mimetic Least Squares 47 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
The sharp, ♯, and flat, ♭, acting on a 2-form (ω2 = wdxdy) and a scalar field (w):
w♭ = wdxdy The case of twisted 0-forms and twisted 2-forms are identical to the corresponding standard forms. For the 2D case, these operations can be summarized by ⋆1 = dxdy, ⋆dx = dy, ⋆dy = −dx ⋆ dxdy = 1 Burke (1985) section 28, or Bossavit (1998) Japanese papers chapter (2):Geometrical
(2):Geometrical objects, Marsden (2002) p.432 and Bossavit (2005), p. 21 and p.23. We can see now that the special form of the 2D De Rham complex in vector form results from converting the usual De Rham complex for 2D in differential form to its vectorial representation using the above relations. The top exact complexes are identical, on the proxies. On the 0-forms:
∂φ ∂x dx + ∂φ ∂y dy ♯ = ∂φ ∂x ex + ∂φ ∂y ey = ∇φ
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On the 1-forms:
∂g ∂x − ∂f ∂y
♯ = ∂f ∂x − ∂g ∂y = ∇ ×
Which is exactly the same. Now, the bottom exact complex in Eq. (??) is identical to the one Eq. (??), on the proxies. On the twisted 0-forms:
φ0♯ = ∂φ ∂x dx + ∂φ ∂y dy ♯ = ∂φ ∂y ex − ∂φ ∂x ey = −∇⊥ ˜ φ0♯ Remembering that the differential of a twisted form is a twisted form. On the twisted 1-forms:
β1♯ = ∂f ∂x + ∂g ∂y
♯ =
w2♯ = ∂f ∂x + ∂g ∂y = ∇ ·
β1♯ where we have used, as before, ˜ β1 = −gdx + fdy.
back Artur Palha and Marc Gerritsma Mimetic Least Squares 49 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work Summary Future work Further reading
back Artur Palha and Marc Gerritsma Mimetic Least Squares 50 / 50