Gradient-Fourier method for nonlinear elliptic partial differential equations in Sobolev space

Lajos Lóczi

Supervisor: János Karátson

Department of Applied Analysis Eötvös Loránd University

June 2000

Contents

1 Introduction 2

2 Solving equations and minimizing functionals 4

3 The gradient method 8

4 Application to nonlinear elliptic partial differential equations 13

5 The Gradient-Fourier method 18

5.1 Convergence estimate of the approximating

sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 The Fourier method for the auxiliary equations . . . . . . . . . . . . . . . 20 5.3 Final convergence estimate of the algorithm . . . . . . . . . . . . . . . . . 22 5.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 1

Introduction

Our thesis deals with the iterative solution of a certain class of nonlinear elliptic partial differential equations. It incorporates two methods: the gradient method and the Fourier method.

The given partial differential equation can be written as F (u) = 0 with a suitable operator F and the unknown function u. The principle of variation will be made use of for the construction of the solution, that is, the solution of the equation is transformed into finding a minimum of an appropriately defined functional Φ using the concept of the potential operator. (The whole idea rests on the fact that if a function attains its minimum at a point, then its derivative vanishes at that point. Here the operator F plays the role of the derivative.) The technical background of this will be discussed in Chapter 2.

In the general case there is no guarantee that the functional Φ has a minimum. Or, if it has, it may have more than one: several local minima can occur. The theorems in Chapter 3 will give sufficient conditions which ensure the existence and uniqueness of a global minimum point, consequently the corresponding equation admits a unique solution.

Then the gradient method will be used to minimize the functional Φ. Another name of the method is the method of steepest descent, which better illustrates the gist of it. One can think of the values of Φ as defining the height of a surface. We pick an initial point on this surface and try to get progressively more accurate approximations to the minimum. A natural way to achieve this is to follow the path of steepest descent on the surface. In other words, we select the direction for the next iteration step to be the one in which the directional derivative—or gradient—of Φ is minimal. This procedure yields an iterative sequence which converges to the minimum point.

In Chapter 4 the gradient method will be applied to the given partial differential equation (or more precisely, to the boundary value problem). The iterative sequence obtained by the gradient method runs in a suitable Hilbert space (namely, in a Sobolev space). Our main reference here is [11].

The construction of the iterative sequence is effectively reduced to the solution of linear boundary value problems of Poisson type. Chapter 5 will show how to tackle these linear partial differential equations using the Fourier method, which is essentially based on Fourier series expansion. At the end, we give an error estimate of the combined Gradient-Fourier method.

Finally we list some further references. The gradient method in general is described in [10, Chapter XV]. The classical gradient method in finite dimensional spaces and its applications can be found in [15] and [17].

There are various other ways to solve nonlinear elliptic problems numerically. The finite element method and the method of finite differences transform the boundary value problem into an algebraic system of equations in ℝ^{N} (see [3, 20]). Another approach is Newton’s method (see for example in [10]), which is also of great practical importance.

The related Hilbert space theory of the finite element method is used to establish error estimates. Iterations using Hilbert space theory are for example in [2], [11], [14] and [18]. Our thesis also employs these techniques.

Acknowledgement

The author^{1} is indebted to János Karátson for his clear and concise lectures, for the invaluable resources and explanations and for his helpful remarks and corrections about the thesis.

1lloczi@cs.elte.hu

Solving equations and minimizing functionals

This chapter introduces some definitions and concepts used in the thesis, and gives the basic tools for later applications. Most of these propositions will not be proved here. (Many of the proofs can be carried out using their counterparts from real analysis.) For an account of the subject, see [10] and [20].

Definition 2.1 Let X and Y be normed spaces, u ∈ X and A: X → Y. We say that the operator A is Gâteaux differentiable at u if

(i) the directional derivative ∂_{v}A(u) ∈ Y exists for all v ∈ X

(ii) X ∋ v ↦ ∂_{v}A(u) ∈ Y is a continuous linear mapping.

The continuous linear mapping in (ii) will be denoted by A′(u). If Y = ℝ (⇒ A′(u) ∈ X^{∗}, the dual space), then instead of A′(u)v we often write ⟨A′(u), v⟩.

The closed line segment between u ∈ X and v ∈ X will be denoted by [u, v], that is [u, v]: = {u + t(v − u) | t ∈ [0, 1]} ⊂ X. (Similarly, the open line segment is (u, v).)

Proposition 2.2 Let X be a normed space, u, v ∈ X and Φ: X → ℝ such that Φ|_{[u,v]} is Gâteaux differentiable. Then there exists ξ ∈ (u, v) such that

Φ(v) − Φ(u) = ⟨Φ′(ξ), v − u⟩ .

Proposition 2.3 Let X be a normed space, u, v ∈ X and Φ: X → ℝ such that Φ|_{[u,v]} is twice Gâteaux differentiable. Then there exists ξ ∈ (u, v) such that

Φ(v) − Φ(u) = ⟨Φ′(u), v − u⟩ + \frac{1}{2} ⟨Φ′′(ξ)(v − u), v − u⟩ .

If X and Y are normed spaces, B(X, Y) will denote the set of continuous linear mappings from X to Y.

Definition 2.4 Let X, Y and Z be normed spaces. An operator A: X → B(Y, Z) is said to be

(i) hemicontinuous, if for all u, v ∈ X and for all w ∈ Y the mapping

ℝ ∋ t ↦ A(u + tv)w ∈ Z

is continuous.

(ii) bihemicontinuous, if for all u, v_{1}, v_{2} ∈ X and for all w ∈ Y the mapping

ℝ^{2} ∋ (s, t) ↦ A(u + sv_{1} + tv_{2})w ∈ Z

is continuous.

Proposition 2.5 Let X be a normed space. Let Φ: X → ℝ be Gâteaux differentiable such that Φ′ is hemicontinuous. Then for all u, v ∈ X

Φ(v) − Φ(u) = ∫_{0}^{1} ⟨Φ′(u + t(v − u)), v − u⟩dt.

Moreover, if Φ is twice Gâteaux differentiable, and Φ′′ is hemicontinuous, then

Φ′(v) − Φ′(u) = ∫_{0}^{1} Φ′′(u + t(v − u))(v − u)dt,

that is, for all h ∈ X

⟨Φ′(v) − Φ′(u), h⟩ = ∫_{0}^{1} ⟨Φ′′(u + t(v − u))(v − u), h⟩dt.

For any functional Φ: X → ℝ and vectors u, v ∈ X, let ϕ_{u,v} denote the following ℝ → ℝ mapping:

ϕ_{u,v}(t): = Φ(u + t(v − u)) (t ∈ [0, 1]).

Definition 2.6 Let X be a normed space. A functional Φ: X → ℝ is said to be (strictly) convex, if for all u, v ∈ X the function ϕ_{u,v}: ℝ → ℝ is (strictly) convex.

Definition 2.7 Let X be a normed space. An operator A: X → X^{∗} is said to be monotone, if for all u, v ∈ X

⟨A(u) − A(v), u − v⟩ ≥ 0.

Proposition 2.8 Let X be a normed space. Let Φ: X → ℝ be twice Gâteaux differentiable. Then the following statements are equivalent:

(i) Φ is convex

(ii) Φ′: X → X^{∗} is monotone

(iii) Φ′′ ≥ 0, that is for all u, h ∈ X ⟨Φ′′(u)h, h⟩ ≥ 0.

The following definitions and statements will establish connection between solving equations and minimizing functionals.

Proposition 2.9 Let X be a normed space and let Φ: X → ℝ be Gâteaux differentiable. If Φ attains its minimum at a point u ∈ X, then Φ′(u) = 0.

Definition 2.10 Let X be a normed space, A: X → X^{∗} and b ∈ X^{∗}. Consider the equation A(u) = b, where u ∈ X is the unknown vector. We say that a functional Φ: X → ℝ is a minimizing functional (with respect to the given equation) if

A(u^{∗}) = b ⟺ Φ(u^{∗}) = min_{X} Φ.

Definition 2.11 Let X be a real normed space. An operator A: X → X^{∗} is said to be a potential operator, if there exists a functional J: X → ℝ such that J is Gâteaux differentiable and J′(u) = A(u) (u ∈ X). (When X is a Hilbert space, the dual of H will be identified with itself.)

Proposition 2.12 Let X be a real normed space and let A: X → X^{∗} be a potential operator with a convex potential J. Let b ∈ X^{∗} arbitrary. Then the following functional

Φ(u): = J(u) − ⟨b, u⟩ ∈ ℝ (u ∈ X)

is a minimizing functional with respect to the equation A(u) = b.

Proposition 2.13 Let X be a real normed space. Let A: X → X^{∗} be Gâteaux differentiable and let A′ be bihemicontinuous. Then the following are equivalent:

(i) A is a potential operator

(ii) for all w ∈ X A′(w) is symmetric (that is ⟨A′(w)u, v⟩ = ⟨A′(w)v, u⟩ , for all u, v ∈ X).

A sufficient condition for the minimum to exist is

Proposition 2.14 Let X be a reflexive normed space. Let Φ: X → ℝ be a Gâteaux differentiable and convex functional, further suppose that

lim_{∥u∥→∞} Φ(u) = ∞.

Then Φ attains its minimum in X. (Moreover, if Φ is strictly convex, then the minimum is unique.)

For the extension theorem we will need

Definition 2.15 Let H be a Hilbert space with a dense subspace D ⊂ H. Let B: D → H be a linear operator with B ≥ λ·I, where λ > 0 is a constant and I is the identity operator. Further, define a scalar product

⟨u, v⟩_{B}: = ⟨Bu, v⟩ (u, v ∈ D).

Then the completion of (D, ⟨·, ·⟩_{B}) is called the energy space (with respect to B), and denoted by H_{B}. The corresponding norm is

∥u∥_{B}: = \sqrt{⟨u, u⟩_{B}}.

Proposition 2.16 Suppose the operator B has the same properties as above. Then the following inequality holds:

∥u∥_{B} ≤ \frac{1}{\sqrt{λ}} · ∥Bu∥ (u ∈ D).

Finally we prove a lemma (see [6]).

Lemma 2.17 Let H be a real Hilbert space. Let F: H → H be the operator defined by

F (u): = u − \frac{2}{M + m}C(u) (u ∈ H),

where m and M (0 < m ≤ M) are constants, C: H → H is a Gâteaux differentiable operator, C′ is hemicontinuous, further C′(u) (u ∈ H) is a bounded self-adjoint operator with

m · ∥u∥^{2} ≤ ⟨C′(u)h, h⟩ ≤ M · ∥u∥^{2} (u, h ∈ H).  (2.1)

Then for u, v ∈ H

∥F (v) − F (u)∥ ≤ \frac{M − m}{M + m} ∥v − u∥ .

Proof. Let u, v ∈ H be fixed. Using Proposition 2.5 we get

F (v) − F (u) = ∫_{0}^{1} F′(u + t(v − u))(v − u)dt =

= v − u − \frac{2}{M + m} · ∫_{0}^{1} C′(u + t(v − u))(v − u)dt.

Define a linear operator L: H → H by

Lx: = x − \frac{2}{M + m} · ∫_{0}^{1} C′(u + t(v − u))xdt (x ∈ H).

Then L is a bounded self-adjoint operator, since C′(w) has these properties for all w ∈ H. Further, using the estimate (2.1)

−\frac{M − m}{M + m} ∥x∥^{2} = ∥x∥^{2} − \frac{2}{M + m}M · ∥x∥^{2} ≤ ⟨Lx, x⟩ ≤

≤ ∥x∥^{2} − \frac{2}{M + m}m · ∥x∥^{2} = \frac{M − m}{M + m} ∥x∥^{2} (x ∈ H).

Hence ∥L∥ ≤ \frac{M−m}{M+m}. Since F (v) − F (u) = L(v − u), the Lemma is proved. ■

The gradient method

The first theorem of this chapter introduces the gradient method. The method here will be used for the iterative solution of equations A(u) = b in real Hilbert spaces. The operator A need not be linear, however, Gâteaux differentiability is required. The proof follows the one described in [12]. The application of the gradient method to continuous linear operators in Hilbert spaces is discussed in [9] and [10]. For the nonlinear case, see [7], [10] and [19]. The most important variant of the method, the conjugate gradient method, can be found in [1] and [5].

Theorem 3.1 Let (H, ⟨·, ·⟩) be a real Hilbert space and A: H → H be a (nonlinear) operator with the following properties:

(i) A is Gâteaux differentiable and A′ is bihemicontinuous

(ii) A′(u) is self-adjoint (u ∈ H)

(iii) there exist functions m, M: [0, ∞) → [0, ∞), m is monotone decreasing, M is monotone increasing and lim_{r→∞} r · m(r) = ∞ such that for all u ∈ H

m(∥u∥) · I ≤ A′(u) ≤ M(∥u∥) · I,

where I is the identity operator.

Then (1) for any b ∈ H there exists a unique u^{∗} ∈ H such that A(u^{∗}) = b

(2) if u_{0} ∈ H is arbitrary, then there exist constants m_{0}, M_{0} (M_{0} ≥ m_{0} > 0) and a constant μ_{0} > 0 such that if

u_{n+1}: = u_{n} − \frac{2}{M_{0} + m_{0}} (A(u_{n}) − b)

then

∥u_{n} − u^{∗}∥ ≤ \frac{1}{μ_{0}} ∥A(u_{0}) − b∥ · (\frac{M_{0} − m_{0}}{M_{0} + m_{0}} )^{n} (n = 0, 1, 2, …)

Proof. First we prove the existence and uniqueness of the solution. By Proposition 2.13,

(i) and (ii) implies that A is a potential operator with potential J: H → ℝ. Let b ∈ H fixed and let Φ(u): = J(u) − ⟨b, u⟩ (u ∈ H). We will show that Φ is strictly convex and lim_{∥u∥→∞} Φ(u) = ∞. In fact,

Φ′(u) = A(u) − b and Φ′′(u) = A′(u) > 0,

thus, by Proposition 2.8, Φ is strictly convex. Using Proposition 2.3, there exists θ ∈ [0, 1] such that

Φ(u) = Φ(0) + Φ′(0)u + \frac{1}{2} ⟨Φ′′(θu)u, u⟩ .

Due to the fact that ∥θu∥ ∈ [0, ∥u∥] and (iii)

Φ(u) ≥ Φ(0) − ∥Φ′(0)∥ · ∥u∥ + \frac{1}{2}m(∥u∥) · ∥u∥^{2} =

= Φ(0) + ∥u∥ (−Φ′(0) + \frac{1}{2}m(∥u∥) · ∥u∥) .

Using the limit condition in (iii), the last term tends to infinity, as ∥u∥ → ∞. Hence—by Proposition 2.14—Φ attains its unique minimum in X. Finally, Proposition 2.12 shows that this Φ is a minimizing functional with respect to the equation A(u) = b.

Now the convergence estimate will be proved. Let u_{0} ∈ H be arbitrary. Because of the limit condition in (iii) again, there exists r_{0} > 0 such that

m(∥u_{0}∥ + r_{0}) · r_{0} > ∥Φ′(u_{0})∥ .

Let B_{0} denote the closed ball in H with centre u_{0} and radius r_{0}. Further, let

m_{0}: = m(∥u_{0}∥ + r_{0}),

M_{0}: = M(∥u_{0}∥ + r_{0}),

t_{0}: = \frac{2}{M_{0} + m_{0}} ,

Q_{0}: = \frac{M_{0} − m_{0}}{M_{0} + M_{0}} .

Then by the definition of u_{n+1}

u_{n+1} = u_{n} − t_{0} · Φ′(u_{n}).

Now we show that if u, v ∈ B_{0} and v = u − t_{0} · Φ′(u), then

∥Φ′(v)∥ ≤ Q_{0} · ∥Φ′(u)∥  (3.1)

The proof goes the same as in Lemma 2.17 with C: = Φ′. It is easy to see that this operator C satisfies all requirements of the Lemma except (2.1). Instead, a weaker version of this condition holds. This time

Lx = x − t_{0} · ∫_{0}^{1} Φ′′(u + t(v − u))xdt

but u + t(v − u) ∈ [u, v] ⊂ B_{0}, and B_{0} is contained in the ball centered on the origin with radius ∥u_{0}∥ + r_{0}. Hence

m_{0} = m(∥u_{0}∥ + r_{0}) ≤ m(∥u + t(v − u)∥)  (3.2)

and similarly

M(∥u + t(v − u)∥) ≤ M(∥u_{0}∥ + r_{0}) = M_{0}.

This implies

m_{0} · ∥x∥^{2} ≤ ⟨Φ′′(u + t(v − u))x, x⟩ ≤ M_{0} · ∥x∥^{2} (x ∈ H)

using (iii). Except for this small modification, the proof of Lemma 2.17 is unchanged. Finally we find that ∥L∥ ≤ \frac{M_{0}−m_{0}}{M_{0}+M_{0}} . Hence Φ′(v) = LΦ′(u) implies ∥Φ′(v)∥ ≤ ∥L∥ · ∥Φ′(u)∥, which completes the proof of (3.1).

Inequality (3.1) will be used to verify that u_{n} ∈ B_{0} (n ∈ ℕ). We prove by induction. For n = 0 it is trivial from the definition of B_{0}. Suppose u_{0}, …, u_{n} ∈ B_{0}. Then

∥u_{n+1} − u_{0}∥ ≤ ∑_{i=0}^{n} ∥u_{i+1} − u_{i}∥ = t_{0} · ∑_{i=0}^{n} ∥Φ′(u_{i})∥ ≤ t_{0} · ∑_{i=0}^{n} Q_{0}^{i} · ∥Φ′(u_{0})∥ ≤

≤ t_{0} · ∥Φ′(u_{0})∥ · \frac{1}{1 − Q_{0}} = ∥Φ′(u_{0})∥ · \frac{2}{M_{0} + m_{0}} · \frac{1}{1 − \frac{M_{0}−m_{0}}{M_{0}+M_{0}}} = \frac{∥Φ′(u_{0})∥}{m_{0}} ≤ r_{0},

that is u_{n+1} ∈ B_{0}.

Finally, since Φ′(u^{∗}) = 0 holds for the unique solution u^{∗}, for n ∈ ℕ we have

⟨Φ′(u_{n}) − Φ′(u^{∗}), u_{n} − u^{∗}⟩ ≤ ∥Φ′(u_{n})∥ · ∥u_{n} − u^{∗}∥ ≤ Q_{0}^{n} · ∥Φ′(u_{0})∥ · ∥u_{n} − u^{∗}∥ .

On the other hand, from Proposition 2.5 we get

⟨Φ′(u_{n}) − Φ′(u^{∗}), u_{n} − u^{∗}⟩ = ∫_{0}^{1} ⟨Φ′′(u^{∗} + t(u_{n} − u^{∗}))(u_{n} − u^{∗}), u_{n} − u^{∗}⟩dt ≥
≥ min {m_{0}, m(∥u^{∗}∥)} · ∥u_{n} − u^{∗}∥^{2} .

(The inequality depends on the same argument as (3.2), due to the fact that u^{∗} + t(u_{n} − u^{∗}) ∈ [u_{n}, u^{∗}], this time by comparing ∥u_{0}∥ + r_{0} and ∥u^{∗}∥.) Let μ_{0}: = min {m_{0}, m(∥u^{∗}∥)}. Hence we conclude

μ_{0} · ∥u_{n} − u^{∗}∥^{2} ≤ Q_{0}^{n} · ∥Φ′(u_{0})∥ · ∥u_{n} − u^{∗}∥ (n ∈ ℕ)

which was to be proved. ■

The next Remark immediately follows from the definition of μ_{0}.

Remark 3.2 In the case of M and m being constant functions,

M_{0}: = M, m_{0}: = m and μ_{0}: = m

is an appropriate choice (and independent of u_{0}).

The criterion of Gâteaux differentiability in this theorem is a rather strong one. Differential operators in general fail to be Gâteaux differentiable, so the above theorem can not be applied to them directly. To overcome the difficulties, we extend the previous result to (nonlinear) operators which themselves are not Gâteaux differentiable but suitably can be transformed into Gâteaux differentiable ones. The gradient method applied to discontinuous linear operators in Hilbert spaces is described in [4] and [10, Chapter XV]. For the nonlinear uniformly monotone case, see [11]; the nonlinear non-uniformly monotone case can be found in [12]. The following theorem is essentially presented there.

Theorem 3.3 Let (H, ⟨·, ·⟩) be a real Hilbert space and D ⊂ H be a dense subspace. Let T: D → H be a nonlinear (and not necessarily continuous) operator. Let B: D → H be a linear operator such that

ran B ⊃ ran T and B ≥ λ · I,

where λ > 0 is a constant and I is the identity operator. Moreover suppose, that

(i) there exists a (nonlinear) operator A: H_{B} → H_{B} such that A|_{D} = B^{−1}T, where H_{B} denotes the corresponding energy space

(ii) A is Gâteaux differentiable and A′ is bihemicontinuous

(iii) A′(u) is self-adjoint (u ∈ H_{B})

(iv) there exist functions m, M: [0, ∞) → [0, ∞), m is monotone decreasing, M is monotone increasing and lim_{r→∞} r · m(r) = ∞ such that for all u, h ∈ H_{B}

m(∥u∥) · ∥h∥^{2} ≤ ⟨A′(u)h, h⟩ ≤ M(∥u∥) · ∥h∥^{2} .

Then (1) for any g ∈ ran B there exists a unique u^{∗} ∈ H_{B} such that

⟨A(u^{∗}), v⟩_{B} = ⟨g, v⟩ (∀v ∈ H_{B}),

where ⟨·, ·⟩_{B} denotes the scalar product in the energy space. (Moreover, if g ∈ ran T, then T (u^{∗}) = g.)

(2) if u_{0} ∈ D is arbitrary, then there exist constants m_{0}, M_{0} (M_{0} ≥ m_{0} > 0) and a constant ν_{0} > 0 such that if

u_{n+1}: = u_{n} − \frac{2}{M_{0} + m_{0}} B^{−1}(T (u_{n}) − g)

then

∥u_{n} − u^{∗}∥_{B} ≤ ν_{0} · ∥ T (u_{0}) − g∥ · (\frac{M_{0} − m_{0}}{M_{0} + m_{0}} )^{n} (n = 0, 1, 2, …)

Proof. Let g ∈ ran B arbitrary. Define b: = B^{−1}g. Then b ∈ D ⊂ H_{B}. Now apply the previous theorem to the operator A: H_{B} → H_{B} and the vector b ∈ H_{B}. (H_{B} plays the role of H in Theorem 3.1.) Hence we get a unique u^{∗} ∈ H_{B} such that A(u^{∗}) = b. This implies

⟨A(u^{∗}), v⟩_{B} = ⟨b, v⟩_{B} = ⟨g, v⟩ (∀v ∈ H_{B}).

One easily checks that if g ∈ ran T, then u^{∗} ∈ D, thus T (u^{∗}) = g.

Now let u_{0} ∈ D be arbitrary. Theorem 3.1 also yields

∥u_{n} − u^{∗}∥_{B} ≤ \frac{1}{μ_{0}} ∥A(u_{0}) − b∥_{B} · (\frac{M_{0} − m_{0}}{M_{0} + m_{0}} )^{n} (n = 0, 1, 2, …),

if

u_{n+1}: = u_{n} − \frac{2}{M_{0} + m_{0}} (A(u_{n}) − b).

It follows by induction that u_{n} ∈ D (n ∈ ℕ). In fact, since ran B^{−1} ⊂ D and A|_{D} = B^{−1}T, then

A(u_{n}) − b = B^{−1}(T (u_{n}) − g) ∈ D,

hence u_{n+1} ∈ D. For the convergence estimate use Proposition 2.16:

∥A(u_{0}) − b∥_{B} = ∥B^{−1}(T (u_{0}) − g)∥_{B} ≤ \frac{1}{\sqrt{λ}} · ∥ T (u_{0}) − g∥ .

This shows that ν_{0}: = \frac{1}{\sqrt{λ}·μ_{0}} is appropriate. ■

Finally, the following Remark is easy to be seen from the definition of ν_{0} and from Remark 3.2.

Remark 3.4 If the functions M and m are constant, then

M_{0}: = M, m_{0}: = m and ν_{0}: = \frac{1}{\sqrt{λ} · m}

can be chosen independently of u_{0}.

Application to nonlinear elliptic partial differential equations

In this chapter the previous results will be applied to a certain class of nonlinear elliptic partial differential equations. (Other sources, where applications of the Hilbert space gradient method to nonlinear elliptic problems are described are [11] (theoretical iteration for the uniformly monotone case), [13] (for the non-uniformly monotone case) and for the numerical iteration see [6].) Now Theorem 3.3 will be used in the following setting.

Let Ω ⊂ ℝ^{N} be a bounded domain which is C^{2}–diffeomorphic to a convex one and let H: = (L^{2}(Ω), ⟨·, ·⟩_{L^{2}(Ω)}), where ⟨f, g⟩_{L^{2}(Ω)} : = ∫_{Ω} fg (f, g ∈ L^{2}(Ω)). (Throughout the thesis L^{2}(Ω) is considered as a real Hilbert space.) We define a differential operator T with domain

dom T: = D: = H^{2}(Ω) ∩ H_{0}^{1}(Ω).

(As usual, H^{2}(Ω) and H_{0}^{1}(Ω) denote the corresponding Sobolev spaces.)

Let

T (u): = −divf(x, ∇u) (u ∈ D),

that is

(T (u))(x): = − ∑_{i=1}^{N} ∂_{x_{i}}[f_{i}(x, ∂_{x_{1}}u(x), …, ∂_{x_{N}} u(x))]

with x = (x_{1}, x_{2}, …, x_{N}) and f = (f_{1}, f_{2}, …, f_{N}), where f_{i}’s are all real-valued functions, satisfying f_{i} ∈ C^{1}(\overline{Ω} × ℝ^{N}). Moreover, suppose that

∂_{p_{j}}f_{i}(x, p) = ∂_{p_{i}}f_{j}(x, p)  (4.1)

for all x ∈ \overline{Ω} and p ∈ ℝ^{N} (i, j = 1, 2, …, N). Assume that there exist constants m and M (0 < m ≤ M) such that

m|ζ|^{2} ≤ ∑_{i,j=1}^{N} ∂_{p_{j}}f_{i}(x, p)ζ_{i}ζ_{j} ≤ M|ζ|^{2}  (4.2)

for any (x, p) ∈ \overline{Ω} × ℝ^{N} and any ζ ∈ ℝ^{N}. (We remark that (4.2) is equivalent to demanding that all eigenvalues of ∂_{p}f(x, p) lie in the interval [m, M], where ∂_{p}f(x, p) is the N × N (symmetric) matrix standing ∂_{p_{j}}f_{i}(x, p) at the (i, j)^{th} position.)

Finally, let

B: = −Δ, dom B: = D.

(Throughout the thesis (−Δ) is considered as an L^{2}(Ω) → L^{2}(Ω) operator with domain H^{2}(Ω) ∩ H_{0}^{1}(Ω).)

The following boundary value problem will be considered:

\begin{matrix}T (u) & = g \\ u|_{∂Ω} & = 0 \end{matrix} \}  (4.3)

where g ∈ L^{2}(Ω) is given and u ∈ D is the unknown function.

First we prove an important lemma (see for example [16]).

Lemma 4.1 The operator (−Δ) in L^{2}(Ω) is symmetric and for any w ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) the following inequality holds:

⟨−Δw, w⟩_{L^{2}(Ω)} ≥ λ_{1} ∥w∥_{L^{2}(Ω)}^{2} ,

where λ_{1} is the smallest eigenvalue of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω).

Proof. Let u, v ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω). Using the Green–formula we have

⟨−Δu, v⟩_{L^{2}(Ω)} = ∫_{Ω} (−Δu)v = ∫_{Ω} ⟨∇u, ∇v⟩_{ℝ^{N}} − ∫_{∂Ω} (−∂_{ν}u)v.

The last integral vanishes since v ∈ H_{0}^{1}(Ω), thus the symmetry of the scalar product in ℝ^{N} implies the symmetry of (−Δ).

Now let λ_{i} and e_{i} (i = 1, 2, …) denote the eigenvalues and eigenfunctions of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω), respectively. It is known that 0 < λ_{1} ≤ λ_{2} ≤ …, and the only accumulation point of the sequence (λ_{i}) is (+∞). Moreover, the eigenfunctions form an orthonormal basis and their linear combinations are dense in L^{2}(Ω) as well as in H_{0}^{1}(Ω). Let a_{i} (i = 1, 2, …) be the coefficients of w ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) in its Fourier series expansion, that is a_{i}: = ∫_{Ω} we_{i}. Then w = ∑_{i=1}^{∞} a_{i}e_{i} in L^{2}(Ω)–norm and ∥w∥_{L^{2}}^{2} = ∑_{i=1}^{∞} a_{i}^{2}. Using the continuity of the scalar product and the symmetry of (−Δ) we have

⟨−Δw, w⟩_{L^{2}} = ⟨−Δw, ∑_{i=1}^{∞} a_{i}e_{i}⟩_{L^{2}} = ∑_{i=1}^{∞} a_{i} ⟨−Δw, e_{i}⟩_{L^{2}} =

= ∑_{i=1}^{∞} a_{i} ⟨w, −Δe_{i}⟩_{L^{2}} = ∑_{i=1}^{∞} a_{i} ⟨w, λ_{i}e_{i}⟩_{L^{2}} = ∑_{i=1}^{∞} a_{i}λ_{i} ⟨w, e_{i}⟩_{L^{2}} = ∑_{i=1}^{∞} λ_{i}a_{i}^{2}.

Since 0 < λ_{1} ≤ λ_{2} ≤ … ≤ λ_{i} ≤ …

∑_{i=1}^{∞} λ_{i}a_{i}^{2} ≥ λ_{1} ∑_{i=1}^{∞} a_{i}^{2} = λ_{1} ∥w∥_{L^{2}}^{2} ,

that is ⟨−Δw, w⟩_{L^{2}} ≥ λ_{1} ∥w∥_{L^{2}}^{2}. ■

Proposition 4.2 The above choices fulfill the assumptions of Theorem 3.3.

Sketch of the proof. It is known that D is a dense subspace in the real Hilbert space L^{2}(Ω). From the previous Lemma we see that the operator B in L^{2}(Ω) and with domain D has a lower bound λ_{1} > 0 (that is B ≥ λ_{1} ·I, where I is the identity operator). Further, ran T ⊂ L^{2}(Ω) and ran B = L^{2}(Ω) (see [8]), hence ran T ⊂ ran B.

Let u ∈ D. Using the definition of the scalar product in the energy space and the Green–formula again (as in the proof of the previous Lemma), there holds

⟨u, u⟩_{B} = ⟨Bu, u⟩_{L^{2}(Ω)} = ∫_{Ω} (−Δu)u = ∫_{Ω} ⟨∇u, ∇u⟩_{ℝ^{N}} = ∫_{Ω} |∇u|^{2},

that is

∥u∥_{B}^{2} : = ⟨u, u⟩_{B} = ∫_{Ω} |∇u|^{2}.  (4.4)

Here the last expression is a norm in H_{0}^{1}(Ω) too, and D ⊂ L^{2}(Ω) is dense, thus H_{B} = H_{0}^{1}(Ω).

For the construction of the operator A observe that if u, v ∈ D, then

∫_{Ω} T (u)v = ∫_{Ω} − ∑_{i=1}^{N} ∂_{x_{i}}[f_{i}(x, ∇u(x))] · v(x)dx =

= ∫_{Ω} ∑_{i=1}^{N} f_{i}(x, ∇u(x)) · ∂_{i}v(x)dx = ∫_{Ω} ⟨f(x, ∇u(x)), ∇v(x)⟩_{ℝ^{N}} dx,

using the divergence theorem. It is easy to see from Cauchy–Scwartz inequality and from (4.4) that for any fixed u ∈ H_{0}^{1}(Ω) the following assignment

v ↦ ∫_{Ω} ⟨f(x, ∇u(x)), ∇v(x)⟩_{ℝ^{N}} dx

defines a bounded linear functional on H_{0}^{1}(Ω). Hence Riesz’s theorem gives a unique vector A(u) ∈ H_{0}^{1}(Ω) for any u ∈ H_{0}^{1}(Ω) such that

⟨A(u), v⟩_{H_{0}^{1}(Ω)} = ∫_{Ω} ⟨f(x, ∇u(x)), ∇v(x)⟩_{ℝ^{N}} dx.  (4.5)

The induced nonlinear operator A: H_{0}^{1}(Ω) → H_{0}^{1}(Ω) has the required property A|_{D} = B^{−1}T, since

⟨A(u), v⟩_{H_{0}^{1}(Ω)} = ⟨T (u), v⟩_{L^{2}(Ω)} = ⟨B^{−1}T (u), v⟩_{B}

for all v ∈ D, which is a dense subspace, and consequently

A(u) = B^{−1}T (u)

for all u ∈ D.

To prove that A is Gâteaux differentiable and A′ is bihemicontinuous, let

∂_{p}f(x, ∇u(x))

denote the N × N matrix in which ∂_{p_{j}}f_{i}(x, ∇u(x)) stands at the (i, j)^{th} position. Let u ∈ H_{0}^{1}(Ω) be arbitrary, then using (4.2) and a similar argument as above we find that the assignment

H_{0}^{1}(Ω) × H_{0}^{1}(Ω) ∋ (h, v) ↦ ∫_{Ω} ⟨∂_{p}f(x, ∇u(x)) ∇h(x), ∇v(x)⟩_{ℝ^{N}} dx ∈ ℝ

is a bounded bilinear functional. Again, Riesz’s theorem proves the existence of an operator S(u): H_{0}^{1}(Ω) → H_{0}^{1}(Ω) (for each u ∈ H_{0}^{1}(Ω)) for which

⟨S(u)h, v⟩_{H_{0}^{1}(Ω)} = ∫_{Ω} ⟨∂_{p}f(x, ∇u(x)) ∇h(x), ∇v(x)⟩_{ℝ^{N}} dx.  (4.6)

Now it can be shown that A′(u) = S(u), that is

\frac{1}{t} ∥A(u + th) − A(u) − tS(u)h∥_{H_{0}^{1}(Ω)} → 0

as t → 0 ∈ ℝ for any fixed u, h ∈ H_{0}^{1}(Ω); or, since the dual space of H_{0}^{1}(Ω) can be identified with itself, it is equivalent to showing that

\frac{1}{t} sup_{∥v∥_{H^{1}_{0}}=1} ⟨A(u + th) − A(u) − tS(u)h, v⟩_{H_{0}^{1}(Ω)} → 0

as t → 0. In fact, writing back (4.5) and (4.6) and using Lagrange’s equality, cancellation by t occurs. Then, due to the continuity assumptions imposed on f_{i}’s, the integrands tend to 0 (almost everywhere) as t → 0. Finally, since the integrands can be majorated by functions belonging to L^{1}(Ω), Lebesgue’s theorem yields the desired result. By the definition of bihemicontinuity we have to show that

∥A′(u + sw_{1} + tw_{2})h − A′(u)h∥_{H_{0}^{1}(Ω)} → 0

as (s, t) → (0, 0) ∈ ℝ^{2} for any fixed u, w_{1}, w_{2}, h ∈ H_{0}^{1}(Ω). This can be proved using a similar reasoning as above. (For details, see for example [11] and [13].)

It follows from (4.1) that ∂_{p}f(x, ∇u(x)) is a symmetric matrix, so (4.6) and A′(u) = S(u) implies that A′(u) is self-adjoint (u ∈ H_{0}^{1}(Ω)).

Finally, the monotone functions can be chosen to be the constants m and M in (4.2), because

m ∥h∥_{H_{0}^{1}(Ω)}^{2} = ∫_{Ω} m|∇h|^{2} ≤ ∫_{Ω} ⟨∂_{p}f(x, ∇u(x)) ∇h(x), ∇h(x)⟩_{ℝ^{N}} dx =

= ⟨A′(u)h, h⟩_{H_{0}^{1}(Ω)} ≤ ∫_{Ω} M|∇h|^{2} = M ∥h∥_{H_{0}^{1}(Ω)}^{2}

for all u, h ∈ H_{0}^{1}(Ω). ■

Now applying Theorem 3.3 and taking into consideration Remark 3.4 we get the following

Corollary 4.3 Consider the boundary value problem

\begin{matrix}T (u) & = g \\ u|_{∂Ω} & = 0 \end{matrix} \}  (4.7)

where T is defined as above. Let g ∈ L^{2}(Ω) be arbitrary. Then the problem (4.7) admits a unique weak solution u^{∗} ∈ H_{0}^{1}(Ω):

⟨A(u^{∗}), v⟩_{H_{0}^{1}(Ω)} = ⟨g, v⟩_{L^{2}(Ω)} , ∀v ∈ H_{0}^{1}(Ω),

that is for all v ∈ H_{0}^{1}(Ω)

∫_{Ω} ⟨f(x, ∇u^{∗}(x)), ∇v(x)⟩_{ℝ^{N}} dx = ∫_{Ω} g(x)v(x)dx

holds. (If u^{∗} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω), then T (u^{∗}) = g.)

Moreover, if u_{0} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) is arbitrary, then the following sequence

u_{n+1}: = u_{n} − \frac{2}{M + m}(−Δ)^{−1}(T (u_{n}) − g) (n ∈ ℕ)

converges to the solution u^{∗}:

∥u_{n} − u^{∗}∥_{H_{0}^{1}(Ω)} ≤ \frac{1}{m\sqrt{λ_{1}}} ∥ T (u_{0}) − g ∥_{L^{2}(Ω)} · (\frac{M − m}{M + m})^{n} ,

where λ_{1} is the smallest eigenvalue of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω). ■

The main advantage of the method so far is that the solution of the nonlinear equation (i.e. determining T^{−1}) is reduced to the solution of a sequence of linear equations: the inverse of (−Δ) is to be calculated in each iteration step, which is—in general—a much easier task.

The Gradient-Fourier method

In the previous chapter we have constructed—using the gradient method—a sequence u_{n} which converges to the unique solution u^{∗}. Obtaining the next term u_{n+1} of the sequence requires a linear Poisson equation to be solved at each step. (These equations will be referred to as auxiliary equations.) There are various ways to deal with Poisson equations. In this thesis we introduce the Gradient-Fourier method: solutions to the auxiliary equations will be determined using the Fourier method.

Formally, solving a Poisson equation is equivalent to applying the inverse of the Laplacian operator to the right hand side of the equation. In the vast majority of cases, however, symbolic computation is not possible, so in general the exact sequence u_{n} is only theoretically known. A numerical method (now the Fourier method) yields an approximation \overline{u}_{n} of the sequence. Then our goal is to guarantee that the limit of the approximating sequence \overline{u}_{n} still lies near the exact solution u^{∗}.

Let us first introduce some notations. For n = 0, 1, …, let

\overline{u}_{0} : = u_{0},
\overline{u}_{n+1} : = \overline{u}_{n} − \frac{2}{M + m}\overline{z}_{n},

where \overline{z}_{n} is the solution to the auxiliary equation obtained by the Fourier method. (As in the previous chapter, homogeneous Dirichlet boundary conditions are imposed on the auxiliary equations.) We also define two other sequences

z_{n}: = (−Δ)^{−1}(T (u_{n}) − g),
z_{n}^{∗}: = (−Δ)^{−1}(T (\overline{u}_{n}) − g),

i.e. z_{n} is the exact solution to the auxiliary equation with exact (but only theoretically known) right hand side, and z_{n}^{∗} is the exact solution to the auxiliary equation with the approximate (i.e. numerically computed) right hand side. Because of the presence of the inverse of Laplacian, these functions are both unknown, only \overline{z}_{n} is available in practice. Finally, let

E_{n}: = ∥ \overline{u}_{n} − u_{n}∥ .

(The unindexed norms are all understood to be H_{0}^{1}(Ω)–norms throughout the chapter.)

5.1 Convergence estimate of the approximating

sequence

In this section the distance of \overline{u}_{n} and u^{∗} will be estimated provided that \overline{z}_{n} and z_{n}^{∗} remain sufficiently close through the iteration process. The next section will discuss how this can be achieved in the actual construction of \overline{z}_{n}.

First we prove two lemmas. (For the first lemma, see [6].) Lemma 5.1 If ∥\overline{z}_{n} − z_{n}^{∗}∥ ≤ δ_{n} for some δ_{n} (n ∈ ℕ), then

E_{n+1} ≤ \frac{M − m}{M + m}E_{n} + \frac{2}{M + m}δ_{n},

where m and M are given in (4.2). Proof. \overline{u}_{n+1} − u_{n+1} = \overline{u}_{n} − \frac{2}{M+m} \overline{z}_{n} − (u_{n} − \frac{2}{M+m}z_{n}) = = \overline{u}_{n} − u_{n} − \frac{2}{M+m}(\overline{z}_{n} − z_{n} + z_{n}^{∗} − z_{n}^{∗}) = \overline{u}_{n} − u_{n} − \frac{2}{M+m}(z_{n}^{∗} − z_{n}) − \frac{2}{M+m}(\overline{z}_{n} − z_{n}^{∗}) = = \overline{u}_{n} − u_{n} − \frac{2}{M+m}[(−Δ)^{−1}(T (\overline{u}_{n})) − (−Δ)^{−1}(T (u_{n}))] − \frac{2}{M+m}(\overline{z}_{n} − z_{n}^{∗}) = = [\overline{u}_{n} − \frac{2}{M+m}(−Δ)^{−1}(T (\overline{u}_{n}))] − [u_{n} − \frac{2}{M+m}(−Δ)^{−1}(T (u_{n}))] − \frac{2}{M+m}(\overline{z}_{n} − z_{n}^{∗}). Now applying Lemma 2.17 with C: = (−Δ)^{−1} ∘ T yields E_{n+1} = ∥\overline{u}_{n+1} − u_{n+1}∥ = ∥F (\overline{u}_{n}) − F (u_{n}) − \frac{2}{M+m}(\overline{z}_{n} − z_{n}^{∗})∥ ≤ ≤ \frac{M−m}{M+m} ∥\overline{u}_{n} − u_{n}∥ + \frac{2}{M+m} ∥\overline{z}_{n} − z_{n}^{∗}∥ ≤ \frac{M−m}{M+m}E_{n} + \frac{2}{M+m}δ_{n}. ■

Lemma 5.2 Let ε > 0 be arbitrary and δ_{n}: = mε (n ∈ ℕ), then

E_{n} ≤ ε.

Proof. If n = 0 then 0 = ∥\overline{u}_{0} − u_{0}∥ = E_{0} ≤ ε. Suppose the statement is true for some n ∈ ℕ. Then by the previous Lemma we have E_{n+1} ≤ \frac{M−m}{M+m}E_{n} + \frac{2}{M+m}δ_{n} ≤ \frac{M−m}{M+m}ε + \frac{2}{M+m}mε = \frac{M+m}{M+m}ε = ε. ■

Corollary 5.3 Let ε > 0. If ∥ \overline{z}_{n} − z_{n}^{∗}∥ ≤ mε (n ∈ ℕ), then

∥ \overline{u}_{n} − u^{∗}∥ ≤ ε + \frac{1}{m\sqrt{λ_{1}}} ∥ T (u_{0}) − g ∥_{L^{2}(Ω)} · (\frac{M − m}{M + m})^{n} ,

where λ_{1} is the smallest eigenvalue of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω). Proof. ∥ \overline{u}_{n} − u^{∗}∥ ≤ ∥ \overline{u}_{n} − u_{n}∥+∥ u_{n} − u^{∗}∥ ≤ E_{n} + \frac{1}{m\sqrt{λ_{1}}} ∥ T (u_{0}) − g ∥_{L^{2}(Ω)} ·(\frac{M−m}{M+m})^{n} ≤ ≤ ε + \frac{1}{m\sqrt{λ_{1}}} ∥ T (u_{0}) − g ∥_{L^{2}(Ω)} · (\frac{M−m}{M+m})^{n} using Corollary 4.3 and the previous Lemma. ■

5.2 The Fourier method for the auxiliary equations

Now let us focus on a single iteration step (i.e. n ∈ ℕ is fixed in the section). The Fourier method implemented here can be found for example in [16].

According to the definition of z_{n}^{∗}, the auxiliary equation takes the form

\begin{matrix}−Δz_{n}^{∗} & = r_{n}^{∗} \\ z_{n}^{∗}|_{∂Ω} & = 0 \end{matrix} \}  (5.1)

with r_{n}^{∗}: = T (\overline{u}_{n}) − g. Instead of r_{n}^{∗}, the right hand side will be replaced by its Fourier series approximation \overline{r}_{n}. The modified equation is

\begin{matrix}−Δ\overline{z}_{n} & = \overline{r}_{n} \\ \overline{z}_{n}|_{∂Ω} & = 0 \end{matrix} \}  (5.2)

which now can be solved by a formula.

As in the previous chapter, let λ_{i} and e_{i} (i = 1, 2, …) denote the eigenvalues and eigenfunctions of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω), respectively.

Let c_{i} (i = 1, 2, …) be the coefficients of r_{n}^{∗} in its Fourier series expansion, that is

c_{i}: = ∫_{Ω} r_{n}^{∗}e_{i}.

(Of course c_{i} = c_{i,n}, but we simply use c_{i} since n is fixed.) Then

r_{n}^{∗} = ∑_{i=1}^{∞} c_{i}e_{i}.

Now we define \overline{r}_{n} as a partial sum of the above infinite series. Let l (= l(n)) be a positive integer and

\overline{r}_{n}: = ∑_{i=1}^{l} c_{i}e_{i}.

Define \overline{z}_{n} as

\overline{z}_{n}: = ∑_{i=1}^{l} \frac{c_{i}}{λ_{i}} e_{i}.

A simple calculation shows that these satisfy (5.2):

−Δ\overline{z}_{n} = ∑_{i=1}^{l} \frac{c_{i}}{λ_{i}} (−Δe_{i}) = ∑_{i=1}^{l} \frac{c_{i}}{λ_{i}} λ_{i}e_{i} = ∑_{i=1}^{l} c_{i}e_{i} = \overline{r}_{n}.

We will now determine how the requirement of Corollary 5.3 can be fulfilled by the appropriate choice of l. The basis for this is that the weak solution to the equation (−Δ)(\overline{z}_{n} − z_{n}^{∗}) = \overline{r}_{n} − r_{n}^{∗} depends continuously on the right hand side, i.e. there exists a constant c_{∗} such that

∥ \overline{z}_{n} − z_{n}^{∗}∥ ≤ c_{∗} ∥ \overline{r}_{n} − r_{n}^{∗}∥_{L^{2}(Ω)} .

A suitable choice for c_{∗} is \frac{1}{\sqrt{λ_{1}}} due to the following

Lemma 5.4 If

\begin{matrix}−Δw & = f \\ w|_{∂Ω} & = 0 \end{matrix} \}

then

∥w∥ ≤ \frac{1}{\sqrt{λ_{1}}} ∥f∥_{L^{2}} .

Proof. If w ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω), then according to the Green–formula

∫_{Ω} (−Δw)w = ∫_{Ω} |∇w|^{2},

that is ∥w∥^{2} = ⟨−Δw, w⟩_{L^{2}}. Using Lemma 4.1 we get

∥w∥^{2} = ⟨−Δw, w⟩_{L^{2}} = ⟨f, w⟩_{L^{2}} ≤ ∥f∥_{L^{2}} ∥w∥_{L^{2}} ≤ ∥f∥_{L^{2}} \frac{1}{\sqrt{λ_{1}}} ∥w∥ ,

thus

∥w∥ ≤ \frac{1}{\sqrt{λ_{1}}} ∥f∥_{L^{2}} ,

which was to be proved. ■

Hence for Corollary 5.3 it is sufficient to guarantee that

∥\overline{r}_{n} − r_{n}^{∗}∥_{L^{2}} ≤ mε\sqrt{λ_{1}}.  (5.3)

Taking into consideration that

∥r_{n}^{∗} − \overline{r}_{n}∥_{L^{2}}^{2} = ∑_{i=l+1}^{∞} |c_{i}|^{2} = ∑_{i=1}^{∞} |c_{i}|^{2} − ∑_{i=1}^{l} |c_{i}|^{2} = ∥r_{n}^{∗}∥_{L^{2}}^{2} − ∥\overline{r}_{n}∥_{L^{2}}^{2}

we see that for (5.3) it is sufficient that

∥r_{n}^{∗}∥_{L^{2}}^{2} − ∑_{i=1}^{l} |c_{i}|^{2} ≤ λ_{1}m^{2}ε^{2}.

Recalling the definition of r_{n}^{∗} we get the following

Corollary 5.5 If l is a positive integer such that

∑_{i=1}^{l} |c_{i}|^{2} ≥ ∥ T (\overline{u}_{n}) − g∥_{L^{2}}^{2} − λ_{1}m^{2}ε^{2},

then

∥ \overline{z}_{n} − z_{n}^{∗}∥ ≤ mε.

5.3 Final convergence estimate of the algorithm

This section summarizes the results of our thesis. The solution to the given boundary value problem is approximated by an iterative sequence \overline{u}_{n}, which is obtained by the gradient method. Computing each term of the sequence is effectively reduced to a series of numerical integrations via Fourier series expansion. In order to realize the algorithm, one needs to know the eigenvalues and eigenfunctions of the operator (−Δ) on H^{2}(Ω)∩H_{0}^{1}(Ω). With this information, exact error control can be given in H_{0}^{1}(Ω)–norm. Formulating our results leads to the following

Theorem 5.6 Let Ω ⊂ ℝ^{N} be the bounded domain and T be the nonlinear elliptic differential operator defined in Chapter 4. The lower and upper bounds of the coefficients of T are denoted by m and M, respectively, as in (4.2). Let g ∈ L^{2}(Ω) be arbitrary. Then the unique weak solution u^{∗} ∈ H_{0}^{1}(Ω) to the boundary value problem

\begin{matrix}T (u) & = g \\ u|_{∂Ω} & = 0 \end{matrix} \}

can be approximated by the following iterative sequence \overline{u}_{n} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω). Let \overline{u}_{0} ∈ H^{2}(Ω) ∩ H_{0}^{1}(Ω) be arbitrary, and for n ∈ ℕ

\overline{u}_{n+1}: = \overline{u}_{n} − \frac{2}{M + m}\overline{z}_{n},

where

\overline{z}_{n}: = ∑_{i=1}^{l(n)} \frac{c_{i,n}}{λ_{i}} e_{i},

and l(n) is a suitable positive integer, further λ_{i} and e_{i} (i = 1, 2, …) denote the eigenvalues and eigenfunctions of (−Δ) on H^{2}(Ω) ∩ H_{0}^{1}(Ω), respectively, and

c_{i,n}: = ∫_{Ω} (T (\overline{u}_{n}) − g) e_{i}.

Let ε > 0 be arbitrary. If l(n) is chosen such that

∑_{i=1}^{l(n)} |c_{i,n}|^{2} ≥ ∥ T (\overline{u}_{n}) − g∥_{L^{2}(Ω)}^{2} − λ_{1}m^{2}ε^{2} (n ∈ ℕ),

then

∥ \overline{u}_{n} − u^{∗}∥_{H_{0}^{1}(Ω)} ≤ ε + \frac{1}{m\sqrt{λ_{1}}} ∥ T (\overline{u}_{0}) − g ∥_{L^{2}(Ω)} · (\frac{M − m}{M + m})^{n} .

Proof. The proof simply follows from applying Corollary 5.5 and Corollary 5.3. ■

5.4 An example

Finally we briefly mention an example, which arises in plasticity theory (for details, see [6]).

Let Ω be the unit square in ℝ^{2}, that is Ω: = [0, 1] × [0, 1]. Define the operator T as

T (u): = −div(\overline{g}(|∇u|^{2})∇u),
u|_{∂Ω} = 0,

where \overline{g} is a given scalar-valued function (the so-called strain-stress function) defined in a finite interval, and u = u(x_{1}, x_{2}) is the unknown function. For example, \overline{g} can have the form

\overline{g}(t): = \frac{constant}{1 + \sqrt{1 − \frac{t}{3}}} ,

if 0 ≤ t ≤ t_{0}, with a suitable t_{0} < 3 and \overline{g}(t): = \overline{g}(t_{0}), if t > t_{0}.

Then—if the right hand side is also given—we can obtain approximate solutions to the boundary value problem using the previous Theorem and the sine-system as eigenfunctions on the unit square.

Bibliography

[1] Axelsson, O., Chronopoulos, A.T., On nonlinear generalized conjugate gradient methods, Numer. Math. 69 (1994), No. 1., pp. 1–15.

[2] Axelsson, O., Karátson, J., Double Sobolev gradient preconditioning for elliptic problems, Report 0016, Dept. Math., Univ. Nijmegen, April 2000.

[3] Ciarlet, Ph., The finite element method for elliptic problems, North-Holland, Amsterdam, 1978.

[4] Czách, L., The steepest descent method for elliptic differential equations (in Russian), C.Sc. thesis, 1955.

[5] Daniel, J.W., The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967), No. 1., pp. 10–26.

[6] Faragó, I., Karátson, J., The gradient-finite element method for elliptic problems, to appear in Comp. Math. Appl.

[7] Gajewski, H. Gröger, K., Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.

[8] Hackbusch, W., Elliptic Differential Equations (Theory and Numerical Treatment), Springer, 1992.

[9] Kantorovich, L. V., On an effective method of the solution of the extremal problem of the quadratic functional (in Russian), Dokl. Akad. Nauk. SSSR, 48 (1945), pp. 483– 487.

[10] Kantorovich, L.V., Akilov, G.P., Functional Analysis, Pergamon Press, 1982.

[11] Karátson, J., The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations, J. Appl. Anal., 3 (1997), No. 2., pp. 205–217.

[12] Karátson, J., Gradient method for non-uniformly convex functionals in Hilbert space, to appear in Pure Math. Appl.

[13] Karátson, J., On the gradient method for some nonlinear elliptic boundary value problems, to appear in Acta Math. Hung.

[14] Neuberger, J.W., Sobolev gradients and differential equations, Lecture Notes in Math., No. 1670, Springer, 1997.

[15] Ortega, J.M., Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, Academic Press, 1970.

[16] Simon, L. Baderko, E., Linear Partial Differential Equations of Second Order (in Hungarian), Tankönyvkiadó, Budapest, 1983.

[17] Stoyan, G., Numerical methods (in Hungarian), ELTE Typotex, 1993.

[18] Stoyan, G., Optimal iterative Stokes solvers in the harmonic Velte subspace, Report 99-07, SFB F013, University Linz, 1999.

[19] Vainberg, M., Variational Method and the Method of Monotone Operators in the Theory of Nonlinear Equations, J. Wiley, New York, 1973.

[20] Zeidler, E., Nonlinear functional analysis and its applications, Springer, 1986.