Wavelets on Regular Surfaces Generated by Stokes Potentials

Wavelets on Regular Surfaces Generated by Stokes Potentials

Carsten Mayer∗1

1 Geomathematics Group, TU Kaiserslautern, P.O. Box 3049, 67663 Kaiserslautern

By means of the limit and jump relations of potential theory with respect to the Stokes equations the framework of a tensorialwavelet approach on a regular (Lyapunov-) surface is established. The setup of a multiresolution analysis is defined byinterpreting the kernel functions of the limit and jump integral operators as scaling functions on the regular surfaces. Thedistance of the parallel surface to the surface under consideration thereby represents the scale level in the scaling function.Tensorial scaling functions and wavelets show space localizing properties. Thus, they can be used to represent vector fieldslocally on a regular surface. Furthermore, these functions can be used as ansatz functions for the discretization of Fredholmintegral equations of the second kind which result from the Stokes boundary-value problems with respect to a regular surface.By this, scaling functions and wavelets enable us to give a multiscale representation of the solution of the Stokes problem.This representation will be demonstrated in a concrete example.

1 Construction of Wavelets

The starting point of our construction principle of wavelets on regular surfaces, which are also known as Lyapunov surfaces, isthe Stokes boundary value problem and its fundamental solution. Let Σ ⊂ R

3 be a regular surface, i.e. a closed and compactsurface containing the origin, being free of double points and possessing a continuous normal field ν pointing into the outerspace Σext. Let f on Σ be given, find a flow field u and a scalar pressure P , both being sufficiently smooth and regular atinfinity such that

∆u −∇P = 0 in Σint (resp. Σext) and(u|Σ = f or (σ[u]|Σ) ν = f on Σ

)(1)

where σ[u] is the stress tensor of the flow field u. The bivariate function u given by

u(x, y) = − 18π

(i

|x − y| +(x − y) ⊗ (x − y)

|x − y|3)

, x, y ∈ R3, x �= y.

is called fundamental solution of the Stokes problem and is also known as Stokeslet. i ∈ R3×3 denotes the identity matrix

and x ⊗ y = (xiyj)i,j=1,...,3 ∈ R3×3 is the tensor product of two vectors x, y ∈ R

3. Based on the fundamental solutionwe now define, for τ, σ > 0 and f ∈ L2(Σ, R3), the single layer potential u1 and the double layer potential u2 by u1(x) =∫Σ u(x, y)f(y) dω(y) and u2(x) =

∫Σ k(x, y)f(y) dω(y) , where the kernel k is given by

k(x, y) = − 34π

(x − y) ⊗ (x − y)|x − y|5 ((x − y) · ν(y)) , x, y ∈ R

3, x �= y.

Using these potentials we can formulate the well-known jump relations of potential theory (see [4]). Here, we just present thejump relation for the double layer potential, i.e. lim

τ→0,τ>0u2(x + τν(x)) − u2(x − τν(x)) = f(x) holds in the sense of the

|| · ||∞−norm and of the || · ||2−norm. Writing out this jump relation explicitly, we get

limτ→0τ>0

(F ∗ Φτ )(x) = limτ→0τ>0

∫Σ

Φτ (x, y)f(y) dω(y) = f(x), x ∈ Σ, f ∈ C(0)(Σ, R3), (2)

with the kernel functions Φτ (x, y), τ ∈ (0,∞), x, y ∈ Σ, given by

Φτ (x, y) = − 34π

[((x + τν(x) − y) ⊗ (x + τν(x) − y)

|x + τν(x) − y|5 ((x + τν(x) − y) · ν(y)))

−(

(x − τν(x) − y) ⊗ (x − τν(x) − y)|x − τν(x) − y|5 ((x − τν(x) − y) · ν(y))

)].

∗ Corresponding author: e-mail: [email protected], Phone: +49 631 205 4585, Fax: +49 631 205 4736

PAMM · Proc. Appl. Math. Mech. 6, 573–574 (2006) / DOI 10.1002/pamm.200610267

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


The family of functionsΦτ , τ ∈ (0,∞), are called Σ−scaling functions. The family of Σ−wavelet functions, Ψτ , τ ∈ (0,∞),is defined by Ψτ (x, y) = −τ d

dτ Φτ (x, y), x, y ∈ Σ. Using the wavelet function Ψ the limit relation in (2) is substituted byan integral relation, i.e.∫ ∞

0

∫Σ

Ψτ ( . , y)f(y) dω(y)1τ

dτ =∫ ∞

0

(Ψτ ∗ f)1τ

dτ = f

holds pointwise, uniformly and in the sense of the || · ||2–norm. The term Ψτ ∗ f is called the wavelet transform of thevector field f at scale τ . Using these scaling functions and wavelets we can define filter operators Pτ = f ∗Φτ , which can beinterpreted as low-pass filters and Rτ = f ∗Ψτ , interpreted as band-pass filters. The corresponding image spaces of L2(Σ, R3)under these operators are denoted by Vτ (Σ), called scale space, and Wτ (Σ), called detail space. The scale spaces fulfill thefundamental properties of a multiresolution analysis, i.e. limτ→0,τ>0 Vτ (Σ) is dense in L2(Σ, R3) and Vτ (Σ) ⊂ L2(Σ, R3),0 < τ < ∞. The inclusion Vτ (Σ) ⊂ Vτ ′(Σ) for τ ′ < τ can, at the moment, only be shown for Σ being a sphere. For thecase the case of the Laplace equation this approach presented above for the construction of scaling functions and wavelets hasbeen discussed in [1] and for the Helmholtz equation is has been studied in [2].

2 Wavelet Solution of the Exterior Stokes Problem of the First Kind

To solve the exterior Stokes problem of the first kind in the exterior of a regular surface Σ (see (1)) we make an ansatz usinga double layer potential with unknown layer function ϕ ∈ C(0)(Σ, R3) plus a linear combination of a Stokeslet u(x, 0) and aRotlet r(x, 0) (see [3, 4]), both located at the origin u(x) =

∫Σ k(x, y)ϕ(y) dω(y) + α(ϕ)u(x, 0) + ω(ϕ)r(x, 0), x ∈ Σext .

Taking the limit x → Σ this results in the Fredholm boundary integral equation of second kind given by

12ϕ(x) +

∫Σ

k(x, y)ϕ(y) dω(y) + α(ϕ)u(x, 0) + ω(ϕ)r(x, 0)︸︷︷︸K:C(0)(Σ,R3)→C(0)(Σ,R3), ϕ �→Kϕ

= f(x), x ∈ Σ .

The operator K is linear and compact. To solve this equation we make the ansatz ϕ(x) =∑M

j=1 Φτ (x , yj)aj , x ∈ Σ,where {yj}j=1,...,M is given set of (equidistributed) points on the regular surface Σ. Finally, we have to obtain the unknowncoefficients {aj}j=1,...,M ⊂ (R3)M by solving the system of equations

M∑j=1

(λΦτ (x, yj) +

(KΦτ (·, yj)

)(x)

)aj = f(x), x ∈ Σ.

This can be done by appropriate methods such as collocation, Galerkin procedure or least square approximation. Openproblems in this field are, for example, to find suitable integration rules on regular surfaces, or to find adequate systems{yj} ⊂ Σ of equidistributed point-sets on regular surfaces.

Fig. 1 Left: Numerical solution of the exterior Stokes problem for a sphere with boundary data f = (−1, 0, 0)T . Arrows indicate the flowfield and the color indicates the pressure difference near the object. Parameters used here are M = 50 and τ = 0.75. Middle: Analyticalsolution of the exterior Stokes problem around a sphere with boundary data f . Right: Numerical solution of the exterior Stokes problem foran ellipsoid with boundary data f = (−1, 0, 0)T . Parameters used in this computation are M = 70 and τ = 0.75.

References

[1] W. Freeden, C. Mayer, Wavelets Generated by Layer Potentials, Appl. Comp. Harm. Ana., 14, 195-237, (2003).[2] W. Freeden, C. Mayer, M.Schreiner, Tree Algorithms in Wavelet Approximations by Helmholtz Potential Operators, Num. Func. Anal.

and Opt., 24, Nr. 7 & 8, 747-782, (2003).[3] O.A. Ladyshenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York London Paris, 1969.[4] H. Power and L.C. Wrobel. Boundary Integral Methods in Fluid Mechanics. Computational Mechanics, 1995.


Section 10 574

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Wavelets on Regular Surfaces Generated by Stokes Potentials