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Elliptic partial differential equations Suitable elements on the sphere Numerical results BPX-type preconditioners for 2nd and 4th order elliptic problems on the sphere Jan Maes 1 Angela Kunoth 2 Adhemar Bultheel 1 1 Department of Computer Science Katholieke Universiteit Leuven 2 Institut für Angewandte Mathematik & Institut für Numerische Simulation Universität Bonn 16. Rhein-Ruhr Workshop, 2006

Elliptic partial differential equations Suitable elements on the sphere Numerical results Outline Elliptic partial differential equations 1 Preconditioning Elliptic equations on the sphere Suitable elements on the sphere 2 Spherical spline spaces C 1 Powell–Sabin elements on the sphere C 0 linear elements on the sphere Numerical results 3 Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results Outline Elliptic partial differential equations 1 Preconditioning Elliptic equations on the sphere Suitable elements on the sphere 2 Spherical spline spaces C 1 Powell–Sabin elements on the sphere C 0 linear elements on the sphere Numerical results 3 Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results Preconditioning Simple example: − u ′′ = f on [ 0 , 1 ] , u ( 0 ) = u ( 1 ) = 0 Weak formulation: v ∈ H 1 � u ′ , v ′ � = � f , v � , 0 ([ 0 , 1 ]) , � v � E := � v ′ , v ′ � Galerkin scheme: φ ( x ) := ( 1 − | x | ) + 2 j / 2 φ ( 2 j · − k ) φ j , k ( x ) := span { φ j , k | k = 1 , . . . , 2 j − 1 } S j := � u ′ j , v ′ � = � f , v � , v ∈ S j

Elliptic partial differential equations Suitable elements on the sphere Numerical results Preconditioning Preconditioning through a change of basis: S n = span { ψ j , k | j = 0 , . . . , n , k ∈ I j } System to solve: � c j , k ψ ′ j , k , ψ ′ � l , m � = � f , ψ l , m � j , k Suppose that 2 � � � � | c j , k | 2 ≤ � � � | c j , k | 2 � � γ c j , k ψ j , k ≤ Γ � � � � j , k j , k j , k � � E then κ ( A ) = O (Γ /γ )

Elliptic partial differential equations Suitable elements on the sphere Numerical results Outline Elliptic partial differential equations 1 Preconditioning Elliptic equations on the sphere Suitable elements on the sphere 2 Spherical spline spaces C 1 Powell–Sabin elements on the sphere C 0 linear elements on the sphere Numerical results 3 Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results Poisson and the biharmonic equation on the 2-sphere Tangential gradient ∇ S u := ∇ u − ( n · ∇ u ) n , n outward normal to S The Laplace–Beltrami operator on the 2-sphere S ∆ S := ∇ S · ∇ S We are interested in: Poisson Biharmonic ∆ 2 − ∆ S u = f on S S u = f on S �∇ S u , ∇ S v � = � f , v � � ∆ S u , ∆ S v � = � f , v �

Elliptic partial differential equations Suitable elements on the sphere Numerical results Outline Elliptic partial differential equations 1 Preconditioning Elliptic equations on the sphere Suitable elements on the sphere 2 Spherical spline spaces C 1 Powell–Sabin elements on the sphere C 0 linear elements on the sphere Numerical results 3 Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Elliptic partial differential equations Suitable elements on the sphere Numerical results Outline Elliptic partial differential equations 1 Preconditioning Elliptic equations on the sphere Suitable elements on the sphere 2 Spherical spline spaces C 1 Powell–Sabin elements on the sphere C 0 linear elements on the sphere Numerical results 3 Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical Powell–Sabin splines PS refinement ∆ PS of ∆ Bézier ordinates Space of spherical PS splines s ( P i ) = f i , D g i s ( P i ) = f gi , D h i s ( P i ) = f hi , ∀ i has a unique solution in S 1 2 (∆ PS )

Elliptic partial differential equations Suitable elements on the sphere Numerical results Spherical PS basis functions Define the spherical B-spline B ij by B ij ( P k ) = δ ik α ij , D g i B ij ( P k ) = δ ik β ij , D h i B ij ( P k ) = δ ik γ ij , ∀ k graph of ( B ij ( v ) + 1 ) v , v ∈ S graph of B ij ( v ) v , v ∈ S

Elliptic partial differential equations Suitable elements on the sphere Numerical results Connection with bivariate PS B-spline Any spherical function f has a homogeneous extension of degree d , i.e. � v � v ∈ R 3 \ { 0 } ( f ) d ( v ) := | v | d f , | v | 1–1 connection with bivariate PS B-spline Let T i be the tangent plane to S at P i . The restriction of � v � 2 ( v ) := | v | 2 B ij � � B ij | v | to T i coincides with a corresponding bivariate PS B-spline

Elliptic partial differential equations Suitable elements on the sphere Numerical results Connection with bivariate PS B-spline Any spherical function f has a homogeneous extension of degree d , i.e. � v � v ∈ R 3 \ { 0 } ( f ) d ( v ) := | v | d f , | v | 1–1 connection with bivariate PS B-spline Let T i be the tangent plane to S at P i . The restriction of � v � 2 ( v ) := | v | 2 B ij � � B ij | v | to T i coincides with a corresponding bivariate PS B-spline

Elliptic partial differential equations Suitable elements on the sphere Numerical results Connection with bivariate PS B-spline ← bivariate B-spline on T i ← sphere ← spherical B-spline

Elliptic partial differential equations Suitable elements on the sphere Numerical results Stability of spherical PS B-spline basis The 1–1 connection is the key ingredient to extend stability results for bivariate B-splines to spherical B-splines ⇓ The spherical PS B-splines form a stable basis with respect to � · � E , provided that the corresponding bivariate PS B-splines form a stable basis with respect to � · � E .

Elliptic partial differential equations Suitable elements on the sphere Numerical results Stability of spherical PS B-spline basis The 1–1 connection is the key ingredient to extend stability results for bivariate B-splines to spherical B-splines ⇓ The spherical PS B-splines form a stable basis with respect to � · � E , provided that the corresponding bivariate PS B-splines form a stable basis with respect to � · � E .