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 of Technology Email: a.palhadasilvaclerigo@tudelft.nl April 12, 2016 Artur Palha and Marc Gerritsma Mimetic Least Squares 1 / 50
Outline The Standard Least Squares Mimetic Approach Summary and Future Work The Standard Least Squares How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work? Mimetic Approach Going back to the basics Differential geometry Mimetic least-squares Summary and Future Work Summary Future work Further reading Artur Palha and Marc Gerritsma Mimetic Least Squares 2 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The Standard Least Squares How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work? Mimetic Approach Going back to the basics Differential geometry Mimetic least-squares Summary and Future Work Summary Future work Further reading Artur Palha and Marc Gerritsma Mimetic Least Squares 3 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The principle The partial differential equation � L u = f in Ω R u = h on Γ Artur Palha and Marc Gerritsma Mimetic Least Squares 4 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The principle The partial differential equation � L u = f in Ω R u = h on Γ Reduce the dimension of the problem (discretize) � L u h,p = f in Ω R u n,p = h on Γ Artur Palha and Marc Gerritsma Mimetic Least Squares 4 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The principle Translate to a minimization problem � � I ( u h,p ; f, h ) ≡ 1 �L u h,p − f � 2 X h , Ω + �R u h,p − h � 2 min X h , Ω 2 u h,p ∈ X h,p Which reduces to: � � � � � � � � L u h,p , L v h,p Ω + R u h,p , R v h,p Γ = f, L v h,p Ω + h, R v h,p Γ Artur Palha and Marc Gerritsma Mimetic Least Squares 5 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The principle Translate to a minimization problem � � I ( u h,p ; f, h ) ≡ 1 �L u h,p − f � 2 X h , Ω + �R u h,p − h � 2 min X h , Ω 2 u h,p ∈ X h,p Which reduces to: � � � � � � � � L u h,p , L v h,p Ω + R u h,p , R v h,p Γ = f, L v h,p Ω + h, R v h,p Γ And finally to an algebraic system Au h,p = b Artur Palha and Marc Gerritsma Mimetic Least Squares 5 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? The finite dimensional spaces: C 0 nodal elements All physical quantities represented by similar spaces � φ i,j h p i ( x ) h p φ ( x, y ) → φ h ( x, y ) = j ( y ) i,j � � m,n h p m ( x ) h p � m,n u x n ( y ) u ( x, y ) → u h ( x, y ) = � k,l u y m,n h p k ( x ) h p l ( y ) That is: � � h p i ( x ) h p φ h ( x, y ) ∈ span j ( y ) , i, j = 0 , . . . , p � � n ( y ) ⊗ h p k ( x ) h p h p m ( x ) h p u h ( x, y ) ∈ span l ( y ) , m, n, k, l = 1 , . . . , p h p i ( ξ ) Lagrange interpolants over Gauss-Lobatto-Legendre points. Artur Palha and Marc Gerritsma Mimetic Least Squares 6 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? Numerical solution of 2D Poisson equation φ ( x, y ) mimetic φ 1.0 1.0 1.8 1.8 0.5 1.5 0.5 1.5 1.2 1.2 0.0 0.0 0.9 0.9 0.6 0.6 0.5 0.5 0.3 0.3 1.0 0.0 1.0 0.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Artur Palha and Marc Gerritsma Mimetic Least Squares 7 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? Numerical solution of 2D Poisson equation v x ( x, y ) mimetic v x mimetic q x 1.0 1.0 1.6 1.6 0.5 1.0 0.5 1.0 0.4 0.4 0.0 0.0 0.2 0.2 0.8 0.8 0.5 0.5 1.4 1.4 1.0 2.0 1.0 2.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Artur Palha and Marc Gerritsma Mimetic Least Squares 8 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? Numerical solution of 2D Poisson equation v y ( x, y ) mimetic v y mimetic q y 1.0 1.0 1.6 1.76 0.5 1.0 0.5 1.10 0.4 0.44 0.0 0.0 0.2 0.22 0.88 0.8 0.5 0.5 1.54 1.4 1.0 2.20 1.0 2.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Artur Palha and Marc Gerritsma Mimetic Least Squares 9 / 50
Outline How does it work? The Standard Least Squares The finite dimensional spaces Mimetic Approach Example: 2D Poisson equation Summary and Future Work Why it does not work? Why it does not work? We are not respecting the structure of the equations in the discrete setting Artur Palha and Marc Gerritsma Mimetic Least Squares 10 / 50
Outline Going back to the basics The Standard Least Squares Differential geometry Mimetic Approach Mimetic least-squares Summary and Future Work The Standard Least Squares How does it work? The finite dimensional spaces Example: 2D Poisson equation Why it does not work? Mimetic Approach Going back to the basics Differential geometry Mimetic least-squares Summary and Future Work Summary Future work Further reading Artur Palha and Marc Gerritsma Mimetic Least Squares 11 / 50
Outline Going back to the basics The Standard Least Squares Differential geometry Mimetic Approach Mimetic least-squares Summary and Future Work Physical quantities and geometry 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 � ∂V D · d A = Q ( V ) � ∇ · B = 0 ∂V B · d A = 0 � − ∂ B − ∂ � ∇ × E = ∂S E · d l = S B · d A ∂t ⇐ ⇒ ∂t � J + ∂D S J · d A + ∂ ∇ × H = ∂S H · d l = � � S D · d A ∂t ∂t = ǫ E = ǫ E D D B = µ H B = µ H Artur Palha and Marc Gerritsma Mimetic Least Squares 12 / 50
Outline Going back to the basics The Standard Least Squares Differential geometry Mimetic Approach Mimetic least-squares Summary and Future Work Classification by orientation 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 on the system. (electric current J , electric induction D , mass flux 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 Going back to the basics The Standard Least Squares Differential geometry Mimetic Approach Mimetic least-squares Summary and Future Work Inner and outer orientation of geometrical objects Artur Palha and Marc Gerritsma Mimetic Least Squares 14 / 50
Outline Going back to the basics The Standard Least Squares Differential geometry Mimetic Approach Mimetic least-squares Summary and Future Work Classification of physical laws Topological laws 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 , . . . Constitutive laws 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 = ǫ E , q = ρ v , . . . Artur Palha and Marc Gerritsma Mimetic Least Squares 15 / 50
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