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Elliptic partial differential equations Suitable elements on the sphere Numerical results

BPX-type preconditioners for 2nd and 4th

• rder elliptic problems on the sphere

Jan Maes1 Angela Kunoth2 Adhemar Bultheel1

1Department of Computer Science

Katholieke Universiteit Leuven

2Institut 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

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Preconditioning

Simple example: −u′′ = f

• n [0, 1],

u(0) = u(1) = 0 Weak formulation: u′, v′ = f, v, v ∈ H1

0([0, 1]),

vE := v′, v′ Galerkin scheme: φ(x) := (1 − |x|)+ φj,k(x) := 2j/2φ(2j · −k) Sj := span{φj,k | k = 1, . . . , 2j − 1} u′

j, v′ = f, v,

v ∈ Sj

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Preconditioning

Preconditioning through a change of basis: Sn = span{ψj,k | j = 0, . . . , n, k ∈ Ij} System to solve:

• j,k

cj,kψ′

j,k, ψ′ l,m = f, ψl,m

Suppose that γ

• j,k

|cj,k|2 ≤

• j,k

cj,kψj,k

• 2

E

≤ Γ

• j,k

|cj,k|2 then κ(A) = O(Γ/γ)

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results 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 ∇Su := ∇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 −∆Su = f on S ∇Su, ∇Sv = f, v Biharmonic ∆2

Su = f on S

∆Su, ∆Sv = f, v

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results 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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(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(αv) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(τ), τ ∈ ∆}

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results 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(Pi) = fi, Dgis(Pi) = fgi, Dhis(Pi) = fhi, ∀i has a unique solution in S1

2(∆PS)

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Spherical PS basis functions

Define the spherical B-spline Bij by Bij(Pk) = δikαij, DgiBij(Pk) = δikβij, DhiBij(Pk) = δikγij, ∀k graph of Bij(v)v, v ∈ S graph of (Bij(v) + 1)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. (f)d (v) := |v|df v |v|

• ,

v ∈ R3 \ {0} 1–1 connection with bivariate PS B-spline Let Ti be the tangent plane to S at Pi. The restriction of

• Bij
• 2 (v) := |v|2Bij

v |v|

• to Ti 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. (f)d (v) := |v|df v |v|

• ,

v ∈ R3 \ {0} 1–1 connection with bivariate PS B-spline Let Ti be the tangent plane to S at Pi. The restriction of

• Bij
• 2 (v) := |v|2Bij

v |v|

• to Ti 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 Ti ← 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.

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results

The spherical spline space S0

1(∆)

This case is much easier. Similar results hold.

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Poisson equation

−∆Su = 2x on S (solution: u = x) We used a hierarchical basis of spherical hat functions

BPX HB n κ residual #iter κ residual #iter 1 3.1 2.4897e-05 12 7.6 2.4974e-05 17 2 3.7 1.6766e-05 9 10.7 1.9546e-05 16 3 4.6 4.7350e-06 11 15.2 8.6198e-06 20 4 5.5 4.5474e-06 11 22.2 5.2361e-06 22 5 6.2 1.6705e-06 12 31.9 3.0622e-06 23 6 6.7 1.0193e-06 12 44.9 1.4750e-06 25 7 7.0 6.2720e-07 12 60.9 6.5043e-07 26 8 7.4 1.6451e-07 13 84.2 3.4960e-07 24

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Outline

1

Elliptic partial differential equations Preconditioning Elliptic equations on the sphere

2

Suitable elements on the sphere Spherical spline spaces C1 Powell–Sabin elements on the sphere C0 linear elements on the sphere

3

Numerical results Poisson equation Biharmonic equation

Elliptic partial differential equations Suitable elements on the sphere Numerical results

Biharmonic equation

∆2

Su = 36xy on S

(solution: u = xy) We used a hierarchical basis of spherical PS B-splines

BPX HB n κ residual #iter κ residual #iter 1 52.0 2.2290e-03 59.1 2.2222e-03 2 66.7 5.1424e-04 2 81.2 3.8158e-04 2 3 78.4 4.2928e-04 1 106.5 6.2316e-04 4 87.7 3.1846e-04 3 144.6 2.5778e-04 5 5 95.2 1.6570e-04 3 199.9 1.5944e-04 14 6 100.6 7.9261e-05 5 274.4 6.8452e-05 4 7 105.5 4.0583e-05 4 375.8 4.2450e-05 15

Elliptic partial differential equations Suitable elements on the sphere Numerical results

References

P . Alfeld, M. Neamtu, and L. L. Schumaker. Bernstein–Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided

• Geom. Design, 13:333–349, 1996.
• W. Dahmen and A. Kunoth. Multilevel preconditioning. Numer. Math.,

63:315–344, 1992.

• G. Dziuk. Finite elements for the Beltrami operator on arbitrary
• surfaces. In Partial differential equations and calculus of variations, S.

Hildebrandt and R. Leis, eds., Lecture Notes in Mathematics 1357, Springer, Berlin, 1988, pp. 142–155.

• J. Maes and A. Bultheel. A hierarchical basis preconditioner for the

biharmonic equation on the sphere. Accepted for publication in IMA J.

• Numer. Anal., 2006.
• J. Maes, A. Kunoth and A. Bultheel. BPX-type preconditioners for 2nd

and 4th order elliptic problems on the sphere. Submitted for publication.