Generating Minimal Surfaces Subject to the

Plateau Problems by Finite Element Method

Hong Gu

Research Institute of Symbolic ComputationJohannes Kepler University, A4232, Linz, Austria

hgu@risc.uni-linz.ac.at

Abstract. There already exists avariety of softwares for generating min-imal surfaces of special types. However, the convergence theories for thoseapproximating methods are always left uncompleted. This leads to thedifficulty of displaying a whole class of minimal surface in general form.In this paper, we discuss the finite element approximating methods to theminimal surfaces which are subject to the well-known Plateau probems.The differential form of the Plateau problems will be given and, for solv-ing the associated discrete scheme, either the numerical Newton iterationmethod can be applied or we can try some promsing symbolic approaches.The convergence property of the numerical solutions is proved and thismethod will be applied to generating the minimal surface graphically oncertain softwares later. The method proposed in this paper has muchlower complexity and fits for inplementing the two grid and parallel al-gorithms to speed up the computation.Key Words: Plateau Problem, Variational form, Convexity, Brouwer’sfixed point theorem.

1 Introduction

The study of minimal surfaces is a branch of differential geometry, because themethods of differential calculus are applied to geometrical problems. One of theoldest questions is: “What is the surface of smallest area spanned by a givencontour?” Such question is nontrivial despite the fact that every physical soapfilm appears to know the answer. The existence of the minimal surface withinsuch a fixed contour is usually taken as the well-known Plateau Problem, whichhas been solved by Rado, Douglas and Osserman in the 1930’s.

From the point of view of local geometry, a minimal surface can also be takenas the surface which has the zero average mean curvature on each surface point,e.g., a saddle shape.

Previously unknown and certainly unexpected minimal surfaces were foundby David Hoffman and his collaborators at GANG, the Center for Geometry,Analysis, Numerics, and Graphics at the University of Massachusetts in 1985

This work has been partially supported by the Austrian Fonds zur Forderung derwissenschaftlichen Forschung (FWF) under project nr. SFB F013/F1304

I. Dimov et al. (Eds.): NMA 2002, LNCS 2542, pp. 471–478, 2003.c© Springer-Verlag Berlin Heidelberg 2003

472 Hong Gu

(see [5] etc.). They first used their MESH computer graphics system to find thesesurfaces, and then later proved their existence with fully rigorous mathematics.This truly excited the minimal surface community and piqued their interest incomputer graphics.

Many problems in optimal geometry require specialized software systemsbecause there is often no explicit parametrization of a desired minimal surface.Although in the past years, lots of softwares have been produced for generatingminimal surfaces graphically (e.g. [1]), there is very few numerical analysisfor their approximation procedures. One possibility of using the finite elementmethod solving minimal surfaces has been proposed in [7] but solving its discreteform is not cheap by the exact method. In this paper we will discuss the finiteelement solution to the Plateau problem based on another discretization form,give the proof of the convergence and then show some typical experiments.

In the next section we will first propose the differential equations to thePlateau problems and define their variational forms in the finite element space.Some error analysis for solving the linear finite element equations will be pre-viewed in Section 3, and we will focus on the proving of the convergence propertyof the discrete solution to the Plateau problem mainly by Brouwer’s fixed pointtheorem in Section 4. In Section 5 we show some experiments of generating theminimal surface on Maple software for testing this new method.

2 Preliminaries

Due to the fact that the minimal surface has the property of having zero averagemean curvature at each surface point, we can define the Plateau problems interms of a partial differential equation, which satisfying some boundary restric-tions.

Let us assume that such minimal surface can be represented explicitly byz = u(x, y) in a 3-d space with the boundary representation

u = f(x, y), (x, y) ∈ ∂Ω,

then u solves the following system of differential equations:−div((1 + u2y)ux, (1 + u2

x)uy) + 6uxyuxuy = 0 in Ω,

u = f(x, y), on ∂Ω,(1)

where Ω is a strictly convex domain such that it is sufficient to assume that thesolution to the previous equation is unique if f(x, y), which satisfies the boundedslope condition (see [9]), is the restriction to ∂Ω of a function in the Sobolevspace W 2

q (Ω), for some q > 2.Set the nonlinear operator

A(u, v) =∫

Ω

[(1 + u2y)uxvx + (1 + u2

x)uyvy + 6uxyuxuyv]dxdy,

Generating Minimal Surfaces Subject to the Plateau Problems 473

where ux, uxy... denote ∂xu, ∂x(∂yu)..., etc. Then the equation (2.1) weaklyequivalent to the following variational form:

Looking for w ∈ H10 (Ω) ∩ H2(Ω) ∩ C1(Ω), such that

A(w + f , v) = 0, in Ω, ∀v ∈ H10 (Ω) ∩ H2(Ω) ∩ C1(Ω), (2)

where f is the “regular” extension of f from ∂Ω to Ω, especially, f ∈ W 2q (Ω), q >

2. (Obviously, w + f = u according to the previous uniqueness assumption.)We now consider the discrete scheme for (2.2). Let Ωh ⊂ Ω be divided by

mesh Th with size h. Sh(Ωh) is the finite element subspace of H10 (Ωh)∩H2(Ωh)∩

C1(Ωh) and Ih is the corresponding interpolation operator on Sh(Ωh) (see [2,3]etc.). It is also assumed that the mesh partition is regular and all the grid nodeson ∂Ωh also located on ∂Ω.

Then the discrete solution wh ∈ Sh(Ω) is defined by

A(wh + f , v) = 0, in ∀v ∈ Sh(Ωh). (3)

and wh + f = uh will be taken as the finite element solution to the Plateauproblems.

The above discrete scheme appears to be a bunch of polynomial equationsso that can be practically solved either by the Newton iteration method (see[6,15] etc.), or by other promising symbolic eliminations. And furthermore, suchdiscrtization form is cheap to be solved out by a lot of exact computations.

For the continuous solution w is unique according to the assumption, wh =w|Ωh

∈ H10 (Ωh) ∩ H2(Ωh) ∩ C1(Ωh) which satisfying

A(wh + f , v) = 0, in ∀v ∈ H10 (Ωh) ∩ H2(Ωh) ∩ C1(Ωh)

also must be unique. Then we could consider the oringinal problem based on acertain convex polygon domain. For convenience we still take wh as w, Ωh as Ω,etc., and will prove that wh convergence to the real solution w when the meshsize h goes to zero in the following sections.

3 Some Error Resolutions for Solving the Linear FiniteElement Equations

Let L(·) be a linear operator defined on Sh(Ω) by

L(u) = −div(a11ux + a21uy, a12ux + a22uy) + b1ux + b2uy + cu,

where aij , bi, i, j = 1, 2, c ≥ 0 are continuous functions satisfying

a11v2x + a22v

2y + (a12 + a21)vxvy ≥ c0(∇v)2 ∀v ∈ Sh(Ω)

for some c0 > 0.According to the maximum value principle, it can be proved that the follow-

ing equation

A′(Phu, v) = (L(Phu), v) = (f, v) ∀v ∈ Sh(Ω) (4)

is uniquely solvable. (see [10,12,13])

474 Hong Gu

Namely, such mapping: Ph : u → Phu, which satisfying

A′(u − Phu, v) = 0 ∀v ∈ Sh(Ω),

is the Galerkin’s projection operator. And we have the following convergencetheorem (see [2,11] etc.)

Theorem 1. For h 1, Phu admits the following estimate

‖u − Phu‖1,∞ ≤ Ch‖u‖2,∞, (5)

for C > 0.

Moreover, we introduce the discrete Green functions as, given z ∈ Ω, theGreen functions gz

h,x, gzh,y ∈ Sh(Ω) satisfying

A′(v, gzh,x) = vx(z), A′(v, gz

h,y) = vy(z) ∀v ∈ Sh(Ω). (6)

It has been proved that (see [14])

‖gzh,x‖1,1 ≤ C|logh|, (7)

‖gzh,y‖1,1 ≤ C|logh|, (8)

for constant C > 0.

4 The Approximation to the Discrete Plateau Problems

The following lemma shows the sufficient and necessary condition of being thefinite element solution to (2.3).

Lemma 1. For any w, wh satisfies (2.2), (2.3) respectively, and v ∈ Sh(Ω),

A(wh + f , v) = A(w + f , v) + A′(w + f ; wh − w, v) + R(w + f , wh + f , v), (9)

where A′(w + f ; ., .) is the bilinear operator defined as (3.1), when we take thecoefficients aij , bi, c, i, j = 1, 2, respectively according to the first argument as

a11 = 1 + u2y, a22 = 1 + u2

x, a12 = a21 = −uyux,

b1 = 3uyuxy − 3uyyux, b2 = 3uxuxy − 3uxxuy,

and c = 0.Then wh ∈ Sh(Ω) solves (2.3) if and only if

A′(w + f ; w − wh, v) = R(w + f , wh + f , v), ∀v ∈ Sh(Ω). (10)

Further more, if ‖wh‖2,∞ ≤ K, then the remainder R satisfying

R(w + f , wh + f , v) ≤ C(K)‖w − wh‖21,∞‖v‖1,1. (11)

Generating Minimal Surfaces Subject to the Plateau Problems 475

Proof: Set G(t) = A(w + f + t(wh − w), v). then it follows from the identity

G(1) = G(0) + Gt(0) +∫ 1

0

Gtt(t)(1 − t)dt.

It is easy to compute by local intergration that

Gt(0) = A′(w + f ; wh − w, v)

where the coefficients of A′ satisfying the given conditions from the lemma. And,by taking

R(w + f , wh + f , v) =∫ 1

0

Gtt(t)(1 − t)dt.

A straightforward calculation shows that

|R(w + f , wh + f , v)|≤ max|Gtt(t)|≤ C(K)‖w − wh‖2

1,∞‖v‖1,1.

Finally, if wh solves (2.3), then G(1) = 0 and complete the proof.Let Ph be the Galerkin projection with respect to the bilinear from A′(w +

f ; ., .), then we have the following theorem:

Theorem 2. For problems of the form (2.2), there exist a constant C > 0, suchthat for mesh Th with its size h 1, the corresponding finite element equation(2.3) has a unique solution wh satisfying

‖wh − Phw‖1,∞ ≤ Ch2|logh|,and thus we have the following error estimation

‖wh − w‖1,∞ ≤ Ch

Proof: Define a nonlinear operator Φ : Sh(Ω) → Sh(Ω) by

A′(w + f , Φ(v) − w, φ) = R(w + f , v + f , φ) ∀φ ∈ Sh(Ω)

It can be proved according to the last section that, Φ is well defined and acontiunous operator, according to the result from Theorem 3.1 we can obtainthat

‖w − Phw‖1,∞ ≤ Ch.

Then define a set B = v ∈ Sh(Ω) : ‖v − Phw‖1,∞ ≤ Ch, C > 0, and by theinverse estimation, there exist C1, C2, C3 > 0, such that

‖v − w‖2,∞ ≤ ‖v − Ihw‖2,∞ + ‖Ihw − w‖2,∞

≤ C1h−1(‖v − Phw‖1,∞ + ‖Ihw − Phw‖1,∞) + C2

≤ C3,

then we get ‖v‖2,∞ is uniformly bounded.

476 Hong Gu

Now we can prove that Φ(B) ⊂ B which means Φ is a contraction operator.In fact, when we substitute φ by the discrete Green Function φ = gz

h,x andφ = gz

h,y into

A′(w + f ; Φ(v) − Phw, φ) = R(w + f , v + f , φ),

according to Lemma 4.1, (3.4) and (3.5), for all v ∈ B, we get

‖Φ(v) − Phw‖1,∞ ≤ C0|logh|‖w − v‖21,∞

≤ 2C0|logh|(‖Phw − v‖21,∞ + ‖w − Phw‖2

1,∞)

≤ 2C0|logh|(C2h2 + C2h2)

≤ Ch,

where C0, C1 > 0, for h 1.By Brouwer’s fixed point theorem, there exist a solution wh ∈ B, such that

Φ(wh) = wh. And according to Lemma 4.1, wh solves (2.3) and satisfies

‖wh − Phw‖1,∞ ≤ Ch2|logh|.

For uniqueness, we consider that uh should still maintains the minimal areaproperty over the domain Ω as (see [4])

uh = minu|∂Ω=fA∗(uh) =∫

Ω

(1 + |∇uh|2)1/2.

By the fact that the operator A∗ is strictly convex (see [7]) and Sh(Ω) is finitedimensional, it then follows that there exists a unique minimizing solution uh.

5 Graphical Examples for Generating Minimal Surface

To illustrate the features of the approximation method proposed from the lastsections, we now show two model examples.

Example 1.For Ω = [0, 1] × [0, 1], we use the piecewise bilinear finite elements with

mesh size h = 1/16. The border curves are the restriction of smooth functionf = 0.25 − (x − 0.5)2. Then the finite element result after 3 iteration steps isdisplayed in Figure 1.

Example 2.This example shows the convergence property on a trival case. By given the

initial border function the restriction of xy(1 − x)(1 − y) on the same domain,we know the exact solution to this Plateau Problem should be a plane z = 0.

Figure 2 shows the maxium error between the numerical solution and theexact one will not be greater than 2 ∗ 10−4 if we use the same grid partition asthe last example. And based on the relatively coarser grid partition by 8×8 and

Generating Minimal Surfaces Subject to the Plateau Problems 477

0.25

0.2

0.15

0.1

0.05

0

0

1

0.8

0.6

0.4

0.2

0

Fig. 1. The approximated minimal surface

4 × 4 meshes, we get the maximum errors are less than 8 ∗ 10−4 and 4 ∗ 10−3,respectively, which coincident with our theoretical estimations in the last section.

Those experimental datas for the computations are obtained from the Solarisoperating system computer galaxy.risc.uni-linz.ac.at with 1152M memory.

0

-5e-05

-0.0001

-0.00015

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

Fig. 2. The shape after computation which also indicate the error

478 Hong Gu

6 Conclusion and Remarks

In this paper, we discussed the finite element method for solving the Plateauproblems on convex domains, and give the error analysis to the whole approachs.The convergence results here are also basically valid for the bilinear finite elementapproximation (see [4]). Such discrete scheme has a relatively lower complexityand fitted for inplementing the 2-grid technique and parallel approachs (see[8,15,16]) in order to speed up the computation. Some proofs of those speedupalgorithms have been finished in [4] and all the numerical experiments here arecarried out by the Maple software.

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